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Applied Numerical Mathematics
Journal Prestige (SJR): 0.93 Citation Impact (citeScore): 1 Number of Followers: 5 Hybrid journal (It can contain Open Access articles) ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274 Published by Elsevier [3161 journals] |
- Operator-compensation methods with mass and energy conservation for
solving the Gross-Pitaevskii equation- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Xiang-Gui Li, Yongyong Cai, Pengde WangAbstractIn this work, operator-compensation based high order methods are presented to solve the Gross-Pitaevskii equation modeling Bose-Einstein condensation (BEC). We begin with the high-order approximation to the Laplacian by the central finite difference method combined with operator-compensation technique, and the Crank-Nicolson and time-splitting methods are then proposed for the time discretization. We show that the Crank-Nicolson operator-compensation method keeps the mass and energy conservation, while the time-splitting operator-compensation method only conserves the mass. Then by extending the high-order operator-compensation methods for the first order derivative, the time-splitting finite difference method is developed to solve the Gross-Pitaevskii equation with an angular momentum rotation term. Numerical results are given to test accuracy and verify the conservation properties.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Xiang-Gui Li, Yongyong Cai, Pengde WangAbstractIn this work, operator-compensation based high order methods are presented to solve the Gross-Pitaevskii equation modeling Bose-Einstein condensation (BEC). We begin with the high-order approximation to the Laplacian by the central finite difference method combined with operator-compensation technique, and the Crank-Nicolson and time-splitting methods are then proposed for the time discretization. We show that the Crank-Nicolson operator-compensation method keeps the mass and energy conservation, while the time-splitting operator-compensation method only conserves the mass. Then by extending the high-order operator-compensation methods for the first order derivative, the time-splitting finite difference method is developed to solve the Gross-Pitaevskii equation with an angular momentum rotation term. Numerical results are given to test accuracy and verify the conservation properties.
- A second-order accurate scheme for two-dimensional space fractional
diffusion equations with time Caputo-Fabrizio fractional derivative- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jiankang Shi, Minghua ChenAbstractWe provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The equation is derived by continuous time random walk model, being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with L2-norm and mathematical induction with l∞ norm). Moreover, the optimal estimates are obtained.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jiankang Shi, Minghua ChenAbstractWe provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The equation is derived by continuous time random walk model, being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with L2-norm and mathematical induction with l∞ norm). Moreover, the optimal estimates are obtained.
- Conservative limiting method for high-order bicompact schemes as applied
to systems of hyperbolic equations- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Michael D. Bragin, Boris V. RogovAbstractIn this work, a new limiting method for bicompact schemes is proposed that preserves them conservative. The method is based upon a finite-element treatment of the bicompact approximation. An analogy between collocation finite-element schemes and bicompact schemes is established. The proposed method is tested on one-dimensional gas dynamics problems that include the Sedov problem, the “peak test” Riemann problem, the Shu–Osher problem, and the “blast wave” problem. Additionally, the method is tested as applied to a two-dimensional problem for the quasilinear Hopf equation. It is shown on these examples that bicompact schemes with conservative limiting are significantly more accurate than hybrid bicompact schemes.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Michael D. Bragin, Boris V. RogovAbstractIn this work, a new limiting method for bicompact schemes is proposed that preserves them conservative. The method is based upon a finite-element treatment of the bicompact approximation. An analogy between collocation finite-element schemes and bicompact schemes is established. The proposed method is tested on one-dimensional gas dynamics problems that include the Sedov problem, the “peak test” Riemann problem, the Shu–Osher problem, and the “blast wave” problem. Additionally, the method is tested as applied to a two-dimensional problem for the quasilinear Hopf equation. It is shown on these examples that bicompact schemes with conservative limiting are significantly more accurate than hybrid bicompact schemes.
- A Tau–like numerical method for solving fractional delay
integro–differential equations- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Sedaghat Shahmorad, M.H. Ostadzad, D. BaleanuAbstractIn this paper, an operational matrix formulation of the Tau method is herein discussed to solve a class of delay fractional integro–differential equations. The approximate solution is sought by using a suitable matrix representation of fractional and delay integrals. An error bound is herein for the first time discussed. Numerical examples show the effectiveness of the method.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Sedaghat Shahmorad, M.H. Ostadzad, D. BaleanuAbstractIn this paper, an operational matrix formulation of the Tau method is herein discussed to solve a class of delay fractional integro–differential equations. The approximate solution is sought by using a suitable matrix representation of fractional and delay integrals. An error bound is herein for the first time discussed. Numerical examples show the effectiveness of the method.
- A class of globally convergent three-term Dai-Liao conjugate gradient
methods- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Shengwei Yao, Qinliang Feng, Lue Li, Jieqiong XuAbstractDue to their simple structure and low memory requirements, conjugate gradient methods are widely used for solving large-scale optimization problems. In this paper, combining Dai-Liao conjugacy condition with a modified symmetric Perry matrix, a class of three-term Dai-Liao conjugate gradient algorithms are proposed. The given methods possess both Dai-Liao conjugacy condition and sufficient descent condition. Meanwhile, the global convergence of the presented algorithms are established under Wolfe line search for general objective functions. Numerical experiments show that the proposed methods are promising.
- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Shengwei Yao, Qinliang Feng, Lue Li, Jieqiong XuAbstractDue to their simple structure and low memory requirements, conjugate gradient methods are widely used for solving large-scale optimization problems. In this paper, combining Dai-Liao conjugacy condition with a modified symmetric Perry matrix, a class of three-term Dai-Liao conjugate gradient algorithms are proposed. The given methods possess both Dai-Liao conjugacy condition and sufficient descent condition. Meanwhile, the global convergence of the presented algorithms are established under Wolfe line search for general objective functions. Numerical experiments show that the proposed methods are promising.
- Numerical solutions of hybrid fuzzy differential equations in a hilbert
space- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): G. Gumah, M. Naser, M. Al-Smadi, S.K.Q. Al-Omari, D. BaleanuAbstractThe main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W22[a,b]⊕W22[a,b] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W22[a,b]⊕W22[a,b]. To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.
- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): G. Gumah, M. Naser, M. Al-Smadi, S.K.Q. Al-Omari, D. BaleanuAbstractThe main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W22[a,b]⊕W22[a,b] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W22[a,b]⊕W22[a,b]. To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.
- Optimized Schwarz and finite element cell-centered method for
heterogeneous anisotropic diffusion problems- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Thanh Hai Ong, Thi-Thao-Phuong HoangAbstractThe paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.
- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Thanh Hai Ong, Thi-Thao-Phuong HoangAbstractThe paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters.
- Efficient implementation of partitioned stiff exponential Runge-Kutta
methods- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Mahesh Narayanamurthi, Adrian SanduAbstractMultiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using exponential Runge-Kutta methods. We propose specific multiphysics implementations of exponential Runge-Kutta methods satisfying stiff order conditions that were developed in [Hochbruck et al., SIAM J. Sci. Comput., 1998] and [Luan and Osterman, J. Comput. Appl. Math., 2014]. We reformulate stiffly–accurate exponential Runge–Kutta methods in a way that naturally allows of the structure of multiphysics systems, and discuss their application to both component and additively partitioned systems. The resulting partitioned exponential methods only compute matrix functions of the Jacobians of individual components, rather than the Jacobian of the full, coupled system. We derive modified formulations of particular methods of order two, three and four, and apply them to solve a partitioned reaction-diffusion problem. The proposed methods retain full order for several partitionings of the discretized problem, including by components and by physical processes.
- Abstract: Publication date: Available online 15 January 2020Source: Applied Numerical MathematicsAuthor(s): Mahesh Narayanamurthi, Adrian SanduAbstractMultiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using exponential Runge-Kutta methods. We propose specific multiphysics implementations of exponential Runge-Kutta methods satisfying stiff order conditions that were developed in [Hochbruck et al., SIAM J. Sci. Comput., 1998] and [Luan and Osterman, J. Comput. Appl. Math., 2014]. We reformulate stiffly–accurate exponential Runge–Kutta methods in a way that naturally allows of the structure of multiphysics systems, and discuss their application to both component and additively partitioned systems. The resulting partitioned exponential methods only compute matrix functions of the Jacobians of individual components, rather than the Jacobian of the full, coupled system. We derive modified formulations of particular methods of order two, three and four, and apply them to solve a partitioned reaction-diffusion problem. The proposed methods retain full order for several partitionings of the discretized problem, including by components and by physical processes.
- Krylov subspace methods for discrete-time algebraic Riccati equations
- Abstract: Publication date: Available online 16 November 2019Source: Applied Numerical MathematicsAuthor(s): Liping Zhang, Hung-Yuan Fan, Eric King-Wah ChuAbstractWe apply the Krylov subspace methods to large-scale discrete-time algebraic Riccati equations. The solvability of the projected algebraic Riccati equation is not assumed but is shown to be inherited from the original equation. Solvability in terms of stabilizability, detectability, stability radius of the associated Hamiltonian matrix and perturbation theory are considered. We pay particular attention to the stabilizing and the positive semi-definite properties of approximate solutions. Illustrative numerical examples are presented.
- Abstract: Publication date: Available online 16 November 2019Source: Applied Numerical MathematicsAuthor(s): Liping Zhang, Hung-Yuan Fan, Eric King-Wah ChuAbstractWe apply the Krylov subspace methods to large-scale discrete-time algebraic Riccati equations. The solvability of the projected algebraic Riccati equation is not assumed but is shown to be inherited from the original equation. Solvability in terms of stabilizability, detectability, stability radius of the associated Hamiltonian matrix and perturbation theory are considered. We pay particular attention to the stabilizing and the positive semi-definite properties of approximate solutions. Illustrative numerical examples are presented.
- A proximal regularized Gauss-Newton-Kaczmarz method and its acceleration
for nonlinear ill-posed problems- Abstract: Publication date: Available online 10 January 2020Source: Applied Numerical MathematicsAuthor(s): Haie Long, Bo Han, Shanshan TongAbstractWe propose and analyse Kaczmarz-type methods that related to proximal algorithms to solve the nonsmooth hybrid regularization models which are derived from collections of N coupled nonlinear operator equations. To begin with, we introduce a proximal regularized Gauss-Newton-Kaczmarz (PRGNK) method which is constructed by combining the Kaczmarz strategy with a proximal regularized Gauss-Newton (PRGN) iteration. Its convergence analysis is presented under appropriate assumptions, and the numerical experiments on large-scale diffuse optical tomography and parameter identification problems indicate that, PRGNK is clearly faster than the PRGN iteration. Moreover, we incorporate a Nesterov-type acceleration scheme into PRGNK in order to further accelerate the convergence, which leads to a so-called accelerated proximal regularized Gauss-Newton-Kaczmarz (APRGNK) method. Based on the discussion for PRGNK, we also establish the convergence analysis of APRGNK. Meanwhile, the numerical simulations explicitly show that APRGNK makes a remarkable acceleration effect compared with its non-accelerated counterpart.
- Abstract: Publication date: Available online 10 January 2020Source: Applied Numerical MathematicsAuthor(s): Haie Long, Bo Han, Shanshan TongAbstractWe propose and analyse Kaczmarz-type methods that related to proximal algorithms to solve the nonsmooth hybrid regularization models which are derived from collections of N coupled nonlinear operator equations. To begin with, we introduce a proximal regularized Gauss-Newton-Kaczmarz (PRGNK) method which is constructed by combining the Kaczmarz strategy with a proximal regularized Gauss-Newton (PRGN) iteration. Its convergence analysis is presented under appropriate assumptions, and the numerical experiments on large-scale diffuse optical tomography and parameter identification problems indicate that, PRGNK is clearly faster than the PRGN iteration. Moreover, we incorporate a Nesterov-type acceleration scheme into PRGNK in order to further accelerate the convergence, which leads to a so-called accelerated proximal regularized Gauss-Newton-Kaczmarz (APRGNK) method. Based on the discussion for PRGNK, we also establish the convergence analysis of APRGNK. Meanwhile, the numerical simulations explicitly show that APRGNK makes a remarkable acceleration effect compared with its non-accelerated counterpart.
- High-order stroboscopic averaging methods for highly oscillatory delay
problems- Abstract: Publication date: Available online 20 November 2019Source: Applied Numerical MathematicsAuthor(s): M.P. Calvo, J.M. Sanz-Serna, Beibei ZhuAbstractWe introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of arbitrarily high accuracy.
- Abstract: Publication date: Available online 20 November 2019Source: Applied Numerical MathematicsAuthor(s): M.P. Calvo, J.M. Sanz-Serna, Beibei ZhuAbstractWe introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of arbitrarily high accuracy.
- A new class of efficient general linear methods for ordinary differential
equations- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): M. Braś, Z. JackiewiczAbstractWe describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.
- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): M. Braś, Z. JackiewiczAbstractWe describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.
- An alternating direction implicit orthogonal spline collocation method for
the two dimensional multi-term time fractional integro-differential
equation- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): Leijie Qiao, Zhibo Wang, Da XuAbstractIn this paper, a new numerical approximation is discussed for the two dimensional multi-term time fractional integro-differential equation. The proposed technique is based on the high order orthogonal spline collocation (OSC) method for the spatial discretization, the classical L1 approximation for the Caputo fractional derivative, and an alternating direction implicit (ADI) method in time, combined with the second order fractional quadrature rule proposed by Lubich to approximate the integral term. Detailed analysis for the optimal error estimate of the proposed scheme is rigorously discussed. Numerical results are listed to support the theoretical analysis.
- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): Leijie Qiao, Zhibo Wang, Da XuAbstractIn this paper, a new numerical approximation is discussed for the two dimensional multi-term time fractional integro-differential equation. The proposed technique is based on the high order orthogonal spline collocation (OSC) method for the spatial discretization, the classical L1 approximation for the Caputo fractional derivative, and an alternating direction implicit (ADI) method in time, combined with the second order fractional quadrature rule proposed by Lubich to approximate the integral term. Detailed analysis for the optimal error estimate of the proposed scheme is rigorously discussed. Numerical results are listed to support the theoretical analysis.
- Discontinuous Galerkin method for the nonlinear Biot's model
- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): Jing Wen, Jian Su, Yinnian He, Hongbin ChenAbstractIn this paper, we consider the quasi-static nonlinear Biot's model in a mixed formulation. Based on the mixed formulation, a linearized fully discrete scheme is obtained, which is constructed by using discontinuous Galerkin methods for the spatial approximation and the first order backward difference formulae (BDF1) for the temporal discretization. First of all, we prove the fully discrete scheme shall be well-posed in proper norms. Then, a priori error estimates for the proposed numerical scheme are derived. Finally, some numerical examples are given to validate the theoretical analysis results by using the proposed numerical scheme to handle the nonlinear Biot's model.
- Abstract: Publication date: Available online 9 January 2020Source: Applied Numerical MathematicsAuthor(s): Jing Wen, Jian Su, Yinnian He, Hongbin ChenAbstractIn this paper, we consider the quasi-static nonlinear Biot's model in a mixed formulation. Based on the mixed formulation, a linearized fully discrete scheme is obtained, which is constructed by using discontinuous Galerkin methods for the spatial approximation and the first order backward difference formulae (BDF1) for the temporal discretization. First of all, we prove the fully discrete scheme shall be well-posed in proper norms. Then, a priori error estimates for the proposed numerical scheme are derived. Finally, some numerical examples are given to validate the theoretical analysis results by using the proposed numerical scheme to handle the nonlinear Biot's model.
- The energy-preserving finite difference methods and their analyses for
system of nonlinear wave equations in two dimensions- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Dingwen Deng, Dong LiangAbstractThe coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Dingwen Deng, Dong LiangAbstractThe coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.
- Stable discretisations of high-order discontinuous Galerkin methods on
equidistant and scattered points- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jan Glaubitz, Philipp ÖffnerAbstractIn this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear L2-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jan Glaubitz, Philipp ÖffnerAbstractIn this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear L2-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.
- Novel fractional time-stepping algorithms for natural convection problems
with variable density- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jilian Wu, Leilei Wei, Xinlong FengAbstractThis work presents novel fractional time-stepping finite element approaches for solving incompressible natural convection problems with variable density. Compared with other projection methods, the main merit of these methods is that we only need to solve one Poisson equation per time step for the pressure, which is computationally more efficient. First-order stability analyses and error estimates are proved and stability analysis of second-order version is established. Numerical tests which confirm the theoretical analyses are provided.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Jilian Wu, Leilei Wei, Xinlong FengAbstractThis work presents novel fractional time-stepping finite element approaches for solving incompressible natural convection problems with variable density. Compared with other projection methods, the main merit of these methods is that we only need to solve one Poisson equation per time step for the pressure, which is computationally more efficient. First-order stability analyses and error estimates are proved and stability analysis of second-order version is established. Numerical tests which confirm the theoretical analyses are provided.
- A spatial fourth-order maximum principle preserving operator splitting
scheme for the multi-dimensional fractional Allen-Cahn equation- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Dongdong He, Kejia Pan, Hongling HuAbstractIn this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of α(1
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Dongdong He, Kejia Pan, Hongling HuAbstractIn this paper, we consider the numerical study for the multi-dimensional fractional-in-space Allen-Cahn equation with homogeneous Dirichlet boundary condition. By utilizing Strang's second-order splitting method, at each time step, the numerical scheme can be split into three sub-steps. The first and third sub-steps give the same ordinary differential equation, where the solutions can be obtained explicitly. While a multi-dimensional linear fractional diffusion equation needs to be solved in the second sub-step, and this is computed by the Crank-Nicolson scheme together with alternating directional implicit (ADI) method. Thus, instead of solving a multidimensional nonlinear problem directly, only a series of one-dimensional linear problems need to be solved, which greatly reduces the computational cost. A fourth-order quasi-compact difference scheme is adopted for the discretization of the space Riesz fractional derivative of α(1
- Existence of solutions of an explicit energy-conserving scheme for a
fractional Klein–Gordon–Zakharov system- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): J.E. Macías-DíazAbstractIn the article Martínez et al. (2019) [7], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): J.E. Macías-DíazAbstractIn the article Martínez et al. (2019) [7], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.
- Solving system of second-order BVPs using a new algorithm based on
reproducing kernel Hilbert space- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Hussein Sahihi, Tofigh Allahviranloo, Saeid AbbasbandyAbstractIn this study, a new and reliable technique based on Reproducing Kernel Method (RKM) is introduced to solve the linear and nonlinear system of second-order boundary value problem. We propose that RKM without Gram-Schmidt orthogonalization process significantly reduces the CPU time and increases the accuracy of the approximate solutions. The convergence theorem and error analysis are also verified. An algorithm is presented to indicate the steps of implementing the proposed scheme. Since the system of second-order boundary value problem is solvable with the numerical commands in Mathematica software, we examined these commands for numerical examples, compared the results with the present method, and demonstrated the higher precision of the present algorithm.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Hussein Sahihi, Tofigh Allahviranloo, Saeid AbbasbandyAbstractIn this study, a new and reliable technique based on Reproducing Kernel Method (RKM) is introduced to solve the linear and nonlinear system of second-order boundary value problem. We propose that RKM without Gram-Schmidt orthogonalization process significantly reduces the CPU time and increases the accuracy of the approximate solutions. The convergence theorem and error analysis are also verified. An algorithm is presented to indicate the steps of implementing the proposed scheme. Since the system of second-order boundary value problem is solvable with the numerical commands in Mathematica software, we examined these commands for numerical examples, compared the results with the present method, and demonstrated the higher precision of the present algorithm.
- Efficient implementation of the ARKN and ERKN integrators for
multi-frequency oscillatory systems with multiple time scales- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Wei Shi, Xinyuan Wu, Kai LiuAbstractIt is known that, a notable feature of both the multi-frequency and multidimensional ARKN (Runge-Kutta-Nyström methods adapted to oscillatory system) and ERKN (extended Runge-Kutta-Nyström) integrators when they are applied to multi-frequency and multidimensional oscillatory system q″+Mq=f(t,q) with multiple time scales is that they exactly integrate the multi-frequency oscillatory homogeneous system q″+Mq=0. With regard to the efficient implementation issues of the integrators, it is significant to calculate efficiently the matrix-valued functions ϕ0(V) and ϕ1(V) which are involved in the two kinds of integrators, where V=h2M and h is a stepsize. In this paper, we pay attention to efficient implementation issues of the multi-frequency and multidimensional ARKN and ERKN integrators which are closely related to the calculations of ϕ0(V) and ϕ1(V). Using the properties of ϕ0(V) and ϕ1(V) and their relations, we present an efficient algorithm to calculate the two matrix-valued functions at lower cost. Two illuminating numerical examples are accompanied and the numerical results show the remarkable efficiency of the algorithm. We also give an essential stability analysis for ARKN and ERKN integrators on the basis of the different approximations to ϕ0(V) and ϕ1(V) which gains an insight into the importance of the calculations of ϕ0(V) and ϕ1(V).
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Wei Shi, Xinyuan Wu, Kai LiuAbstractIt is known that, a notable feature of both the multi-frequency and multidimensional ARKN (Runge-Kutta-Nyström methods adapted to oscillatory system) and ERKN (extended Runge-Kutta-Nyström) integrators when they are applied to multi-frequency and multidimensional oscillatory system q″+Mq=f(t,q) with multiple time scales is that they exactly integrate the multi-frequency oscillatory homogeneous system q″+Mq=0. With regard to the efficient implementation issues of the integrators, it is significant to calculate efficiently the matrix-valued functions ϕ0(V) and ϕ1(V) which are involved in the two kinds of integrators, where V=h2M and h is a stepsize. In this paper, we pay attention to efficient implementation issues of the multi-frequency and multidimensional ARKN and ERKN integrators which are closely related to the calculations of ϕ0(V) and ϕ1(V). Using the properties of ϕ0(V) and ϕ1(V) and their relations, we present an efficient algorithm to calculate the two matrix-valued functions at lower cost. Two illuminating numerical examples are accompanied and the numerical results show the remarkable efficiency of the algorithm. We also give an essential stability analysis for ARKN and ERKN integrators on the basis of the different approximations to ϕ0(V) and ϕ1(V) which gains an insight into the importance of the calculations of ϕ0(V) and ϕ1(V).
- Frequency-explicit convergence analysis of collocation methods for highly
oscillatory Volterra integral equations with weak singularities- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Junjie Ma, Hongchao KangAbstractFilon collocation methods are well known for their good performances in solving the second-kind highly oscillatory Volterra integral equations with weak singularities. As the frequency increases, these methods can provide much more accurate solutions for oscillatory integral equations. This paper is devoted to investigating frequency-related properties of collocation solutions of highly oscillatory Volterra equations with weakly singular kernels. Optimal convergence rates are obtained by analyzing the error equations and studying asymptotic properties of oscillatory integrals involving Bessel functions. Moreover, numerical experiments are conducted to verify the given theoretical results.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Junjie Ma, Hongchao KangAbstractFilon collocation methods are well known for their good performances in solving the second-kind highly oscillatory Volterra integral equations with weak singularities. As the frequency increases, these methods can provide much more accurate solutions for oscillatory integral equations. This paper is devoted to investigating frequency-related properties of collocation solutions of highly oscillatory Volterra equations with weakly singular kernels. Optimal convergence rates are obtained by analyzing the error equations and studying asymptotic properties of oscillatory integrals involving Bessel functions. Moreover, numerical experiments are conducted to verify the given theoretical results.
- Arbitrarily high-order energy-preserving schemes for the Camassa-Holm
equation- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Chaolong Jiang, Yushun Wang, Yuezheng GongAbstractIn this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. The new system is then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, a symplectic Runge-Kutta method such as the Gauss collocation method is applied for the resulting semi-discrete system to arrive at an arbitrarily high-order fully discrete scheme. We prove that the obtained schemes can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.
- Abstract: Publication date: May 2020Source: Applied Numerical Mathematics, Volume 151Author(s): Chaolong Jiang, Yushun Wang, Yuezheng GongAbstractIn this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. The new system is then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, a symplectic Runge-Kutta method such as the Gauss collocation method is applied for the resulting semi-discrete system to arrive at an arbitrarily high-order fully discrete scheme. We prove that the obtained schemes can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.
- hp-version collocation method for a class of nonlinear Volterra integral
equations of the first kind- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Khadijeh Nedaiasl, Raziyeh Dehbozorgi, Khosrow MaleknejadAbstractIn this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of hp-version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the L2-norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Khadijeh Nedaiasl, Raziyeh Dehbozorgi, Khosrow MaleknejadAbstractIn this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of hp-version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the L2-norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results.
- A note on the optimal degree of the weak gradient of the stabilizer free
weak Galerkin finite element method- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Ahmed Al-Taweel, Xiaoshen WangAbstractRecently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement. The idea is to raise the degree of polynomials j for computing weak gradient. It is shown that if j≥j0 for some j0, then SFWG achieves the optimal rate of convergence. However, large j will cause some numerical difficulties. To improve the efficiency of SFWG and avoid numerical locking, in this note, we provide the optimal j0 with rigorous mathematical proof.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Ahmed Al-Taweel, Xiaoshen WangAbstractRecently, a new stabilizer free weak Galerkin method (SFWG) is proposed, which is easier to implement. The idea is to raise the degree of polynomials j for computing weak gradient. It is shown that if j≥j0 for some j0, then SFWG achieves the optimal rate of convergence. However, large j will cause some numerical difficulties. To improve the efficiency of SFWG and avoid numerical locking, in this note, we provide the optimal j0 with rigorous mathematical proof.
- Numerical analysis of a 4th-order time parallel algorithm for the
time-dependent Navier-Stokes equations- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Wenjia Liu, Yanren Hou, Dandan XueAbstractThis work studies the stability and convergency properties for a time parallel algorithm: RIDC4 with reduced stencils, namely reduced 4th revisionist integral deferred correction method. For the spatial approximation of the velocity and the pressure, a finite element space discretization is applied. We proved that the scheme is almost unconditionally stable and is of 4th-order accuracy in time. Finally, numerical experiments are carried out to verify its time and space convergence orders and the formidable parallel efficiency demonstrated by the time speed-up compared with the classical Backward Euler (BE) methods. In conclusion, within almost same time to BE, the (reduced) RIDC4 scheme could achieve 4th order accuracy in time direction, provided four cores are available.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Wenjia Liu, Yanren Hou, Dandan XueAbstractThis work studies the stability and convergency properties for a time parallel algorithm: RIDC4 with reduced stencils, namely reduced 4th revisionist integral deferred correction method. For the spatial approximation of the velocity and the pressure, a finite element space discretization is applied. We proved that the scheme is almost unconditionally stable and is of 4th-order accuracy in time. Finally, numerical experiments are carried out to verify its time and space convergence orders and the formidable parallel efficiency demonstrated by the time speed-up compared with the classical Backward Euler (BE) methods. In conclusion, within almost same time to BE, the (reduced) RIDC4 scheme could achieve 4th order accuracy in time direction, provided four cores are available.
- Analysis of the stabilized element free Galerkin approximations to the
Stokes equations- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Maryam Kamranian, Mehdi Tatari, Mehdi DehghanAbstractThis paper introduces an element free Galerkin method for solving Stokes equations based on the velocity-pressure formulation. A consistently stabilized element free Galerkin method is proposed and Nitsche's method is applied. We formulate and study the optimal convergence of this method. For this purpose, the concept of the vector-valued moving least squares approximation is employed to derive the error estimation in Sobolev spaces. Finally we report some numerical experiments to illustrate the accuracy of the proposed method.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Maryam Kamranian, Mehdi Tatari, Mehdi DehghanAbstractThis paper introduces an element free Galerkin method for solving Stokes equations based on the velocity-pressure formulation. A consistently stabilized element free Galerkin method is proposed and Nitsche's method is applied. We formulate and study the optimal convergence of this method. For this purpose, the concept of the vector-valued moving least squares approximation is employed to derive the error estimation in Sobolev spaces. Finally we report some numerical experiments to illustrate the accuracy of the proposed method.
- Solution of multi-dimensional Fredholm equations using Legendre scaling
functions- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Harendra Singh, D. Baleanu, H.M. Srivastava, Hemen Dutta, Navin Kumar JhaAbstractIn this article, we construct approximate solution to multi-dimensional Fredholm integral equations of second kind using n-dimensional Legendre scaling functions. Error analysis of the problem is provided in the L2 sense. It is shown that our numerical method is numerically stable. Some examples are discussed based on proposed method to show the importance and accuracy of the proposed numerical method.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Harendra Singh, D. Baleanu, H.M. Srivastava, Hemen Dutta, Navin Kumar JhaAbstractIn this article, we construct approximate solution to multi-dimensional Fredholm integral equations of second kind using n-dimensional Legendre scaling functions. Error analysis of the problem is provided in the L2 sense. It is shown that our numerical method is numerically stable. Some examples are discussed based on proposed method to show the importance and accuracy of the proposed numerical method.
- Efficient computation of highly oscillatory Fourier-type integrals with
monomial phase functions and Jacobi-type singularities- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): A. Ihsan HascelikAbstractWe present a method for fast computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities.It is shown that on an appropriately chosen contour in the complex plane the original integral can be expressed as the sum of two nonoscillatory and rapidly decaying integrals, each of which can be efficiently computed by the generalized Gauss-Laguerre rule or by the double exponential method. For moderate and large frequencies, the proposed method is stable and more efficient than the existing methods. We also give a Mathematica program for the implementation of the method on a computer. The effectiveness and accuracy is tested by means of numerical examples.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): A. Ihsan HascelikAbstractWe present a method for fast computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities.It is shown that on an appropriately chosen contour in the complex plane the original integral can be expressed as the sum of two nonoscillatory and rapidly decaying integrals, each of which can be efficiently computed by the generalized Gauss-Laguerre rule or by the double exponential method. For moderate and large frequencies, the proposed method is stable and more efficient than the existing methods. We also give a Mathematica program for the implementation of the method on a computer. The effectiveness and accuracy is tested by means of numerical examples.
- Numerical construction of spherical t-designs by Barzilai-Borwein
method- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Congpei An, Yuchen XiaoAbstractA point set XN on the unit sphere is a spherical t-design is equivalent to the nonnegative quantity AN,t+1 vanished. We show that if XN is a stationary point set of AN,t+1 and the minimal singular value of basis matrix is positive, then XN is a spherical t-design. Moreover, the numerical construction of spherical t-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical t-designs with N=(t+2)2 points for t+1 up to 127.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Congpei An, Yuchen XiaoAbstractA point set XN on the unit sphere is a spherical t-design is equivalent to the nonnegative quantity AN,t+1 vanished. We show that if XN is a stationary point set of AN,t+1 and the minimal singular value of basis matrix is positive, then XN is a spherical t-design. Moreover, the numerical construction of spherical t-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical t-designs with N=(t+2)2 points for t+1 up to 127.
- Investigation of the Oldroyd model as a generalized incompressible
Navier–Stokes equation via the interpolating stabilized element free
Galerkin technique- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Mostafa Abbaszadeh, Mehdi DehghanAbstractThe main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Mostafa Abbaszadeh, Mehdi DehghanAbstractThe main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results.
- Multi-hp adaptive discontinuous Galerkin methods for simplified P
N
approximations of 3D radiative transfer in non-gray media- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Stefano Giani, Mohammed SeaidAbstractIn this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified PN approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified PN models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified PN approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified PN approximations of radiative transfer in non-gray media.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Stefano Giani, Mohammed SeaidAbstractIn this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified PN approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified PN models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified PN approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified PN approximations of radiative transfer in non-gray media.
- Biorthogonal Rosenbrock-Krylov time discretization methods
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Ross Glandon, Paul Tranquilli, Adrian SanduAbstractMany scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Ross Glandon, Paul Tranquilli, Adrian SanduAbstractMany scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.
- On analysis of kernel collocation methods for spherical PDEs
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Davoud MirzaeiAbstractIn this paper the error analysis of the kernel collocation method for partial differential equations on the unit sphere is presented. A simple analysis is given when the true solutions lie in arbitrary Sobolev spaces. This also extends the previous studies for true solutions outside the associated native spaces. Finally, some experimental results support the theoretical error bounds.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Davoud MirzaeiAbstractIn this paper the error analysis of the kernel collocation method for partial differential equations on the unit sphere is presented. A simple analysis is given when the true solutions lie in arbitrary Sobolev spaces. This also extends the previous studies for true solutions outside the associated native spaces. Finally, some experimental results support the theoretical error bounds.
- Numerical analysis for the Cahn-Hilliard-Hele-Shaw system with variable
mobility and logarithmic Flory-Huggins potential- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Yayu Guo, Hongen Jia, Jichun Li, Ming LiAbstractIn this paper, a fully discrete scheme is proposed for solving the Cahn-Hilliard-Hele-Shaw system with a logarithmic potential and concentration dependent mobility. Using the regularization procedure, the domain for the logarithmic free energy density function F(ϕ) is extended from (−1,1) to (−∞,∞). A convex-splitting method of the energy is adopted in time and the mixed finite element method is used in space. Furthermore, we prove the stability of the proposed numerical method and establish the optimal error estimate in the energy norm. Finally, numerical results are presented to support our theoretical analysis.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Yayu Guo, Hongen Jia, Jichun Li, Ming LiAbstractIn this paper, a fully discrete scheme is proposed for solving the Cahn-Hilliard-Hele-Shaw system with a logarithmic potential and concentration dependent mobility. Using the regularization procedure, the domain for the logarithmic free energy density function F(ϕ) is extended from (−1,1) to (−∞,∞). A convex-splitting method of the energy is adopted in time and the mixed finite element method is used in space. Furthermore, we prove the stability of the proposed numerical method and establish the optimal error estimate in the energy norm. Finally, numerical results are presented to support our theoretical analysis.
- Numerical approximation of the dynamic Koiter's model for the hyperbolic
parabolic shell- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Xiaoqin Shen, Qian Yang, Linjin Li, Zhiming Gao, Tiantian WangAbstractIn this paper, the dynamic Koiter's model is discussed for the hyperbolic parabolic shell, i.e., the saddle surface as its middle surface. The numerical computation for this kind of shell is the most difficult as its complex differential geometry. We provide a numerical algorithm for solving the dynamic problems with finite elements and finite difference methods. At the same time, we do the numerical experiments for the hyperbolic parabolic shell. Concretly, we consider the roof of the Valencia Aquarium as a hyperbolic parabolic shell with a geometric model of the roof deduced. Then, we simulate the deformation of the roof by the dynamic applied force coupled with the dynamic Koiter's model. Finally, a comparison of the displacement distribution on the middle surface is made, which verifies that the dynamic Koiter's model is efficient and that numerical algorithm is stable for the hyperbolic parabolic shell. This result provides a strong theoretical and applicable support for the design and protection of such large-span buildings whose roofs and walls are of one whole object.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Xiaoqin Shen, Qian Yang, Linjin Li, Zhiming Gao, Tiantian WangAbstractIn this paper, the dynamic Koiter's model is discussed for the hyperbolic parabolic shell, i.e., the saddle surface as its middle surface. The numerical computation for this kind of shell is the most difficult as its complex differential geometry. We provide a numerical algorithm for solving the dynamic problems with finite elements and finite difference methods. At the same time, we do the numerical experiments for the hyperbolic parabolic shell. Concretly, we consider the roof of the Valencia Aquarium as a hyperbolic parabolic shell with a geometric model of the roof deduced. Then, we simulate the deformation of the roof by the dynamic applied force coupled with the dynamic Koiter's model. Finally, a comparison of the displacement distribution on the middle surface is made, which verifies that the dynamic Koiter's model is efficient and that numerical algorithm is stable for the hyperbolic parabolic shell. This result provides a strong theoretical and applicable support for the design and protection of such large-span buildings whose roofs and walls are of one whole object.
- Extended block boundary value methods for neutral equations with piecewise
constant argument- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Chengjian Zhang, Cui Li, Jingyao JiangAbstractThis paper is concerned with numerical analysis and computation for neutral equations with piecewise constant argument (NEPCA). The block boundary value methods (BBVMs) are extended to solve NEPCA. Under the conditions of preconsistency and p-order consistency, it is proved that the convergence order of the extended BBVMs can arrive at p. Moreover, a sufficient and necessary condition of the numerical asymptotical stability is derived for linear NEPCA. The presented numerical examples further testify the computational effectiveness of the methods and the obtained theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Chengjian Zhang, Cui Li, Jingyao JiangAbstractThis paper is concerned with numerical analysis and computation for neutral equations with piecewise constant argument (NEPCA). The block boundary value methods (BBVMs) are extended to solve NEPCA. Under the conditions of preconsistency and p-order consistency, it is proved that the convergence order of the extended BBVMs can arrive at p. Moreover, a sufficient and necessary condition of the numerical asymptotical stability is derived for linear NEPCA. The presented numerical examples further testify the computational effectiveness of the methods and the obtained theoretical results.
- Multiscale computational method for nonlinear heat transmission problem in
periodic porous media- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): A. Chakib, A. Hadri, A. Nachaoui, M. NachaouiAbstractThis paper deals with multiscale analysis, by using the correctors, of a nonlinear heat transmission problem in a periodic microscopic structure with multiple components. The occurring nonlinearity is related to the flux condition at the interface between the two parts of the medium. In this work, we present first the multiscale asymptotic expansion of the solution for this kind of problems and the strong convergence of the asymptotic expansion to the solution of the multiscale model. Then, we introduce a numerical algorithm based on the multiscale method for solving this problem. Finally, in order to confirm the efficiency of the proposed algorithm, some numerical results obtained through finite element approximations are presented.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): A. Chakib, A. Hadri, A. Nachaoui, M. NachaouiAbstractThis paper deals with multiscale analysis, by using the correctors, of a nonlinear heat transmission problem in a periodic microscopic structure with multiple components. The occurring nonlinearity is related to the flux condition at the interface between the two parts of the medium. In this work, we present first the multiscale asymptotic expansion of the solution for this kind of problems and the strong convergence of the asymptotic expansion to the solution of the multiscale model. Then, we introduce a numerical algorithm based on the multiscale method for solving this problem. Finally, in order to confirm the efficiency of the proposed algorithm, some numerical results obtained through finite element approximations are presented.
- Maximum error estimates of a MAC scheme for Stokes equations with
Dirichlet boundary conditions- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Haixia Dong, Wenjun Ying, Jiwei ZhangAbstractThis paper presents a rigorous convergence analysis in ℓ2-norm and maximum norm for the Marker-And-Cell (MAC) scheme for solving the incompressible Stokes equations with Dirichlet boundary conditions in two dimensions. The numerical scheme is based on a finite-difference approximation on a staggered grid, where the computational boundary conditions are introduced by reflecting boundary conditions. This scheme is well known to numerically produce second order accuracy for velocity, pressure as well as the gradient of velocity. However, the reflecting boundary conditions lead to the fact that the truncation errors near the boundary is the order of O(1), which presents great difficulties in error analysis to obtain optimal order. In this work, by means of a discrete LBB condition and some auxiliary functions that satisfy discrete Stokes equations to a high-order accuracy, we first prove that the convergence is second order in the ℓ2-norm for velocity, pressure and the gradient of the velocity. After that, according to the property of discrete Green functions, we further provide rigorous analysis to show second-order accuracy in the maximum norm for both the velocity and its gradient. The extension of our results to three dimensional problems would be routine. Numerical results are also provided to demonstrate our theoretical analysis.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Haixia Dong, Wenjun Ying, Jiwei ZhangAbstractThis paper presents a rigorous convergence analysis in ℓ2-norm and maximum norm for the Marker-And-Cell (MAC) scheme for solving the incompressible Stokes equations with Dirichlet boundary conditions in two dimensions. The numerical scheme is based on a finite-difference approximation on a staggered grid, where the computational boundary conditions are introduced by reflecting boundary conditions. This scheme is well known to numerically produce second order accuracy for velocity, pressure as well as the gradient of velocity. However, the reflecting boundary conditions lead to the fact that the truncation errors near the boundary is the order of O(1), which presents great difficulties in error analysis to obtain optimal order. In this work, by means of a discrete LBB condition and some auxiliary functions that satisfy discrete Stokes equations to a high-order accuracy, we first prove that the convergence is second order in the ℓ2-norm for velocity, pressure and the gradient of the velocity. After that, according to the property of discrete Green functions, we further provide rigorous analysis to show second-order accuracy in the maximum norm for both the velocity and its gradient. The extension of our results to three dimensional problems would be routine. Numerical results are also provided to demonstrate our theoretical analysis.
- Collocation method for approximate solution of Volterra
integro-differential equations of the third-kind- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): F. Shayanfard, H. Laeli Dastjerdi, F.M. Maalek GhainiAbstractIn this paper, we have adapted the collocation method for solving a special class of Volterra integro-differential equations of the third-kind. Structure of the method and its applicability for proposed equations have been presented. Moreover we have investigated the solvability and convergence analysis of the method. Also some numerical examples are provided to illustrate the theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): F. Shayanfard, H. Laeli Dastjerdi, F.M. Maalek GhainiAbstractIn this paper, we have adapted the collocation method for solving a special class of Volterra integro-differential equations of the third-kind. Structure of the method and its applicability for proposed equations have been presented. Moreover we have investigated the solvability and convergence analysis of the method. Also some numerical examples are provided to illustrate the theoretical results.
- A multi-step scheme based on cubic spline for solving backward stochastic
differential equations- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Long Teng, Aleksandr Lapitckii, Michael GüntherAbstractA multi-step scheme on time-space grids for solving backward stochastic differential equations was proposed in a previous paper, where Lagrange interpolating polynomials are used to approximate the time-integrands with given values of these integrands at chosen multiple time levels. However, the number of time levels is limited, as the stability is lost if the number of time levels is large. For a better stability and the admission of more time levels, in this work, the application of spline replacing Lagrange interpolating polynomials is investigated to approximate the time-integrands. The resulting scheme is a semi-discretization in the time direction involving conditional expectations, which can be numerically solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for arbitrarily many multiple time levels without loosing stability. Several numerical examples including applications in finance are presented to show better approximation properties of this adapted scheme.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Long Teng, Aleksandr Lapitckii, Michael GüntherAbstractA multi-step scheme on time-space grids for solving backward stochastic differential equations was proposed in a previous paper, where Lagrange interpolating polynomials are used to approximate the time-integrands with given values of these integrands at chosen multiple time levels. However, the number of time levels is limited, as the stability is lost if the number of time levels is large. For a better stability and the admission of more time levels, in this work, the application of spline replacing Lagrange interpolating polynomials is investigated to approximate the time-integrands. The resulting scheme is a semi-discretization in the time direction involving conditional expectations, which can be numerically solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Our new proposed multi-step scheme allows for arbitrarily many multiple time levels without loosing stability. Several numerical examples including applications in finance are presented to show better approximation properties of this adapted scheme.
- The time discontinuous space-time finite element method for fractional
diffusion-wave equation- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Yunying Zheng, Zhengang ZhaoAbstractIn this paper a time discontinuous space-time finite element method for fractional diffusion-wave equation is studied. The existence and the uniqueness of the numerical solution are included. When we adopt the shape function as the piecewise r-order function, the convergence order is proved to be r+1 in the sense of L2 norm and r+1/2 in the sense of max norm, respectively. Numerical examples with comparisons the accuracy and effectiveness of proposed method are presented.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Yunying Zheng, Zhengang ZhaoAbstractIn this paper a time discontinuous space-time finite element method for fractional diffusion-wave equation is studied. The existence and the uniqueness of the numerical solution are included. When we adopt the shape function as the piecewise r-order function, the convergence order is proved to be r+1 in the sense of L2 norm and r+1/2 in the sense of max norm, respectively. Numerical examples with comparisons the accuracy and effectiveness of proposed method are presented.
- Optimal error estimate of elliptic problems with Dirac sources for
discontinuous and enriched Galerkin methods- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Sanghyun Lee, Woocheol ChoiAbstractWe present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Sanghyun Lee, Woocheol ChoiAbstractWe present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results.
- A rational approximation of the sinc function based on sampling and the
Fourier transforms- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Sanjar M. Abrarov, Brendan M. QuineAbstractIn our previous publications we have introduced the cosine product-to-sum identity [17]∏m=1Mcos(t2m)=12M−1∑m=12M−1cos(2m−12Mt) and applied it for sampling [1], [2] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy ∼10−14. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Sanjar M. Abrarov, Brendan M. QuineAbstractIn our previous publications we have introduced the cosine product-to-sum identity [17]∏m=1Mcos(t2m)=12M−1∑m=12M−1cos(2m−12Mt) and applied it for sampling [1], [2] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only 16 summation terms can provide accuracy ∼10−14. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented.
- On the hexagonal Shepard method
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Francesco Dell'Accio, Filomena Di TommasoAbstractThe problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Francesco Dell'Accio, Filomena Di TommasoAbstractThe problem of Lagrange interpolation of functions of two variables by quadratic polynomials based on nodes which are vertices of a triangulation has been recently studied and local six-tuples of vertices which assure the uniqueness and the optimal-order of the interpolation polynomial are known. Following the idea of Little and the theoretical results on the approximation order and accuracy of the triangular Shepard method, we introduce an hexagonal Shepard operator with quadratic precision and cubic approximation order for the classical problem of scattered data approximation without least square fit.
- Unconditional superconvergence analysis of a two-grid finite element
method for nonlinear wave equations- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Dongyang Shi, Ran WangAbstractThis paper aims to present a two-grid method (TGM) with low order nonconforming EQ1rot finite element for solving a class of nonlinear wave equations, and to give the superconvergent error analysis unconditionally. Firstly, for the Crank-Nicolson fully discrete scheme, the existence and uniqueness of the numerical solutions are proved, and based on the special characters of this element as well as the priori estimates, the supercloseness and superconvergence analysis of both the original variable u and the auxiliary variable q=ut in the broken H1-norm are deduced on the coarse mesh for the Galerkin finite element method (FEM). Then, by employing the interpolation postprocessing approach and the boundness of the numerical solution in the broken H1-norm on the coarse mesh, the corresponding superconvergence results of order O(τ2+H4+h2) for the TGM are obtained unconditionally, here τ, H and h denote time step, coarse and fine grid sizes, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Dongyang Shi, Ran WangAbstractThis paper aims to present a two-grid method (TGM) with low order nonconforming EQ1rot finite element for solving a class of nonlinear wave equations, and to give the superconvergent error analysis unconditionally. Firstly, for the Crank-Nicolson fully discrete scheme, the existence and uniqueness of the numerical solutions are proved, and based on the special characters of this element as well as the priori estimates, the supercloseness and superconvergence analysis of both the original variable u and the auxiliary variable q=ut in the broken H1-norm are deduced on the coarse mesh for the Galerkin finite element method (FEM). Then, by employing the interpolation postprocessing approach and the boundness of the numerical solution in the broken H1-norm on the coarse mesh, the corresponding superconvergence results of order O(τ2+H4+h2) for the TGM are obtained unconditionally, here τ, H and h denote time step, coarse and fine grid sizes, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.
- Compensated θ-Milstein methods for stochastic differential equations with
Poisson jumps- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Quanwei Ren, Hongjiong TianAbstractIn this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated θ-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated θ-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated θ-Milstein methods with different θ values. Numerical results are given to verify our theoretical results.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Quanwei Ren, Hongjiong TianAbstractIn this paper, we are concerned with numerical methods for solving stochastic differential equations with Poisson-driven jumps. We construct a class of compensated θ-Milstein methods and study their mean-square convergence and asymptotic mean-square stability. Sufficient and necessary conditions for the asymptotic mean-square stability of the compensated θ-Milstein methods when applied to a scalar linear test equation are derived. We compare the asymptotic mean-square stability region of the linear test equation with that of the compensated θ-Milstein methods with different θ values. Numerical results are given to verify our theoretical results.
- Long-term analysis of stochastic θ-methods for damped
stochastic oscillators- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Vincenzo Citro, Raffaele D'AmbrosioAbstractWe analyze long-term properties of stochastic θ-methods for damped linear stochastic oscillators. The presented a-priori analysis of the error in the correlation matrix allows to infer the long-time behaviour of stochastic θ-methods and their capability to reproduce the same long-term features of the continuous dynamics. The theoretical analysis is also supported by a selection of numerical experiments.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Vincenzo Citro, Raffaele D'AmbrosioAbstractWe analyze long-term properties of stochastic θ-methods for damped linear stochastic oscillators. The presented a-priori analysis of the error in the correlation matrix allows to infer the long-time behaviour of stochastic θ-methods and their capability to reproduce the same long-term features of the continuous dynamics. The theoretical analysis is also supported by a selection of numerical experiments.
- Error propagation analysis for a static convergent beam triple LIDAR
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Theodore C. Holtom, Anthony C. BroomsAbstractWe consider the matter of how to assess uncertainty propagation for the converging beam triple LIDAR technology, which is used for measuring wind velocity passing through a fixed point in space. Converging beam triple LIDAR employs the use of three non-parallel, non-coplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order to determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. One aim of this paper is to present the detailed formulae that would be required for the computation of the estimated variances of the reconstructed velocities, calculated on the basis of the error propagation formulae (and to highlight to the practitioner the conditions under which such estimated variances could be relied upon). The other main aim of this paper is to demonstrate that for (various wind profiles and, in particular) certain LIDAR configurations, which can be characterized by an associated parallelepiped with unit edge length, the estimated variances have the potential to be unsuitably large, thus indicating that the reconstructed velocity may be unreliable for gauging the value of the true wind velocity and that the practitioner should avoid such configurations in the measurement campaign.
- Abstract: Publication date: April 2020Source: Applied Numerical Mathematics, Volume 150Author(s): Theodore C. Holtom, Anthony C. BroomsAbstractWe consider the matter of how to assess uncertainty propagation for the converging beam triple LIDAR technology, which is used for measuring wind velocity passing through a fixed point in space. Converging beam triple LIDAR employs the use of three non-parallel, non-coplanar, laser beams which are directed from a fixed platform, typically at ground level, that extend to meet at the point at which measurement of velocity is sought. Coordinate values of the velocity are ascertained with respect to unit vectors along the lines of sight of the laser beams (Doppler vectors), which are then resolved in order to determine the velocity in terms of Cartesian coordinates (i.e. with respect to the standard basis). However, if there is any discrepancy between the recorded values of the coordinates with respect to the Doppler unit vectors and/or the perceived angle settings for such vectors with what they really should be, however small, then this will lead to errors in the reconstructed Cartesian coordinates. One aim of this paper is to present the detailed formulae that would be required for the computation of the estimated variances of the reconstructed velocities, calculated on the basis of the error propagation formulae (and to highlight to the practitioner the conditions under which such estimated variances could be relied upon). The other main aim of this paper is to demonstrate that for (various wind profiles and, in particular) certain LIDAR configurations, which can be characterized by an associated parallelepiped with unit edge length, the estimated variances have the potential to be unsuitably large, thus indicating that the reconstructed velocity may be unreliable for gauging the value of the true wind velocity and that the practitioner should avoid such configurations in the measurement campaign.
- Efficient and energy stable method for the Cahn-Hilliard phase-field model
for Diblock copolymers- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Jun Zhang, Chuanjun Chen, Xiaofeng YangAbstractIn this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Jun Zhang, Chuanjun Chen, Xiaofeng YangAbstractIn this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically.
- A novel iterative method for discrete Helmholtz decomposition
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): JaEun Ku, Lothar ReichelAbstractA new iterative method for the computation of the discrete Helmholtz decomposition of a vector is presented. We are particularly interested in computing the discrete Helmholtz decomposition when the given vector is discretized by a mixed finite element method defined by Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) elements. The decomposition is computed by solving a system of linear equations by an iterative method, that splits a given vector into a divergence-free component and a curl-free component. Each iteration cycle uses a well-developed solver based on the algebraic multigrid method for computing a projection onto H(div) or H(curl). Only a few iteration cycles are required to compute an accurate approximate solution. As a by-product, we obtain an iterative method for the solution of linear systems of equations with a nearly singular matrix.
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): JaEun Ku, Lothar ReichelAbstractA new iterative method for the computation of the discrete Helmholtz decomposition of a vector is presented. We are particularly interested in computing the discrete Helmholtz decomposition when the given vector is discretized by a mixed finite element method defined by Raviart-Thomas (RT) or Brezzi-Douglas-Marini (BDM) elements. The decomposition is computed by solving a system of linear equations by an iterative method, that splits a given vector into a divergence-free component and a curl-free component. Each iteration cycle uses a well-developed solver based on the algebraic multigrid method for computing a projection onto H(div) or H(curl). Only a few iteration cycles are required to compute an accurate approximate solution. As a by-product, we obtain an iterative method for the solution of linear systems of equations with a nearly singular matrix.
- Convergence and superconvergence analysis of finite element methods for
the time fractional diffusion equation- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Meng Li, Dongyang Shi, Lifang PeiAbstractIn this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2-1σ method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Grönwall inequalities, the theoretical analysis including the L2-norm error estimates and H1-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H1-norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis.
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): Meng Li, Dongyang Shi, Lifang PeiAbstractIn this paper, two classes of finite element methods (FEMs) for the two-dimensional time fractional diffusion equation (TFDE) with non-smooth solution are proposed and analyzed. In the temporal direction, we adopt nonuniform L2-1σ method, and in the spatial direction, the conforming bilinear element and the nonconforming modified quasi-Wilson element are utilized. For the proposed conforming and nonconforming FEMs, by using new fractional discrete Grönwall inequalities, the theoretical analysis including the L2-norm error estimates and H1-norm superclose results are given in details. Furthermore, by virtue of the interpolated postprocessing techniques, the global H1-norm superconvergence results are presented. Finally, some numerical results illustrate the correctness of theoretical analysis.
- A new mathematical model for traveling sand dunes: analysis and
approximation- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): M. Falcone, S. Finzi VitaAbstractWe present a new two-layer closed form model for the dynamics of desert dunes under the effect of a horizontal wind blowing in an arbitrary direction. This model is an extension of a very simplified model previously introduced by Hadeler and Kuttler [12]. Our extension, inspired by the sandpile dynamics approach, includes the effects of gravity on both sides (upwind and downwind) of the dune, and allows to describe erosion and deposition in a more accurate way. After a discussion of the model and its properties we present a numerical scheme based on finite differences in 1D and we prove its consistency and stability. Some numerical tests show a good qualitative behavior and a realistic shape for the evolving dunes. Finally, we discuss the preliminary steps of a possible extension of this model to the 2D case.
- Abstract: Publication date: Available online 7 January 2020Source: Applied Numerical MathematicsAuthor(s): M. Falcone, S. Finzi VitaAbstractWe present a new two-layer closed form model for the dynamics of desert dunes under the effect of a horizontal wind blowing in an arbitrary direction. This model is an extension of a very simplified model previously introduced by Hadeler and Kuttler [12]. Our extension, inspired by the sandpile dynamics approach, includes the effects of gravity on both sides (upwind and downwind) of the dune, and allows to describe erosion and deposition in a more accurate way. After a discussion of the model and its properties we present a numerical scheme based on finite differences in 1D and we prove its consistency and stability. Some numerical tests show a good qualitative behavior and a realistic shape for the evolving dunes. Finally, we discuss the preliminary steps of a possible extension of this model to the 2D case.
- The convergence and stability analysis of a numerical method for solving a
mathematical model of language competition- Abstract: Publication date: Available online 31 December 2019Source: Applied Numerical MathematicsAuthor(s): M.R. Eslahchi, Sakine EsmailiAbstractIn this study, a combination of spectral and fixed point methods is applied to solve a mathematical model of language competition. The considered model has formulated the competition between two languages in which the population density of speakers of each language is modelled by a nonlinear parabolic equation. Due to the fact that the problem is nonlinear, the fixed point method is employed to change the problem to a linear one in each step of iterations. After that, the coupled parabolic equations are solved using the spectral method in each step. It is proven that the constructed sequence converges to the exact solution of the problem, which results in the convergence of the method. Then, by applying the method for solving the perturbed problem, the stability of method is proven. By considering some examples the efficiency of the method is demonstrated. Finally, by the use of an example, the influences of various factors on the evolution of the population densities are illustrated step by step.
- Abstract: Publication date: Available online 31 December 2019Source: Applied Numerical MathematicsAuthor(s): M.R. Eslahchi, Sakine EsmailiAbstractIn this study, a combination of spectral and fixed point methods is applied to solve a mathematical model of language competition. The considered model has formulated the competition between two languages in which the population density of speakers of each language is modelled by a nonlinear parabolic equation. Due to the fact that the problem is nonlinear, the fixed point method is employed to change the problem to a linear one in each step of iterations. After that, the coupled parabolic equations are solved using the spectral method in each step. It is proven that the constructed sequence converges to the exact solution of the problem, which results in the convergence of the method. Then, by applying the method for solving the perturbed problem, the stability of method is proven. By considering some examples the efficiency of the method is demonstrated. Finally, by the use of an example, the influences of various factors on the evolution of the population densities are illustrated step by step.
- GeCo: Geometric Conservative nonstandard schemes for biochemical
systems- Abstract: Publication date: Available online 19 December 2019Source: Applied Numerical MathematicsAuthor(s): Angela Martiradonna, Gianpiero Colonna, Fasma DieleAbstractWe generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.
- Abstract: Publication date: Available online 19 December 2019Source: Applied Numerical MathematicsAuthor(s): Angela Martiradonna, Gianpiero Colonna, Fasma DieleAbstractWe generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.
- Time two-grid algorithm based on finite difference method for
two-dimensional nonlinear fractional evolution equations- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): Da Xu, Jing Guo, Wenlin QiuAbstractIn this paper, a time two-grid finite difference (FD) algorithm is proposed for solving two-dimensional nonlinear fractional evolution equations. In this time two-grid FD algorithm, a nonlinear FD system is solved on the time coarse grid of size τC. And the Lagrange's linear interpolation formula is applied to provide some useful values for the time fine grid of size τF. Then, a linear system is solved on the time fine grid. In the temporal direction, a backward Euler method is employed for the time derivative and a first order convolution quadrature rule is applied to discretize the fractional integral term. And the second order central difference quotient is considered for the spatial approximation. By means of the discrete energy method, we obtain the unconditional discrete L2 stability and convergence of order O(τC2+τF+hx2+hy2), where hx and hy are the spatial step sizes in the x direction and the y direction, respectively. Numerical examples are presented to show the feasibility and efficiency of the time two-grid FD algorithm.
- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): Da Xu, Jing Guo, Wenlin QiuAbstractIn this paper, a time two-grid finite difference (FD) algorithm is proposed for solving two-dimensional nonlinear fractional evolution equations. In this time two-grid FD algorithm, a nonlinear FD system is solved on the time coarse grid of size τC. And the Lagrange's linear interpolation formula is applied to provide some useful values for the time fine grid of size τF. Then, a linear system is solved on the time fine grid. In the temporal direction, a backward Euler method is employed for the time derivative and a first order convolution quadrature rule is applied to discretize the fractional integral term. And the second order central difference quotient is considered for the spatial approximation. By means of the discrete energy method, we obtain the unconditional discrete L2 stability and convergence of order O(τC2+τF+hx2+hy2), where hx and hy are the spatial step sizes in the x direction and the y direction, respectively. Numerical examples are presented to show the feasibility and efficiency of the time two-grid FD algorithm.
- Distributed Lagrange multiplier/fictitious domain finite element method
for Stokes/parabolic interface problems with jump coefficients- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): Pengtao Sun, Cheng WangAbstractIn this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD)–mixed finite element method is developed and analyzed for a type of linearized fluid-structure interaction problem–the transient Stokes/parabolic interface problem with jump coefficients. Stability and optimal convergence properties are obtained for both semi- and fully discrete mixed finite element scheme. Numerical experiments are carried out to validate the theoretical results. Similar DLM/FD method can be developed to tackle FSI problems with different compressibility for the fluid and the structure in the future.
- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): Pengtao Sun, Cheng WangAbstractIn this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD)–mixed finite element method is developed and analyzed for a type of linearized fluid-structure interaction problem–the transient Stokes/parabolic interface problem with jump coefficients. Stability and optimal convergence properties are obtained for both semi- and fully discrete mixed finite element scheme. Numerical experiments are carried out to validate the theoretical results. Similar DLM/FD method can be developed to tackle FSI problems with different compressibility for the fluid and the structure in the future.
- Structure preserving model order reduction of a class of second-order
descriptor systems via balanced truncation- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): M. Monir UddinAbstractLarge sparse second-order index-3 descriptor system arises in various disciplines of science and engineering including constraint mechanics, mechatronics (where mechanical and electrical elements are coupled) and circuit designs. Simulation, controller design and design optimization are some applications of such models. These tasks become challenging when the dimension of the system is high. This paper discusses a method to obtain a reduced second-order model from a large sparse second-order index-3 system using the Balanced Truncation. For this purpose, the low-rank alternating direction implicit iteration is modified to solve the Lyapunov equations of the index-3 structure system efficiently in an implicit way. Numerical resultants are discussed to show the reflectivity and efficiency of the techniques.
- Abstract: Publication date: Available online 16 December 2019Source: Applied Numerical MathematicsAuthor(s): M. Monir UddinAbstractLarge sparse second-order index-3 descriptor system arises in various disciplines of science and engineering including constraint mechanics, mechatronics (where mechanical and electrical elements are coupled) and circuit designs. Simulation, controller design and design optimization are some applications of such models. These tasks become challenging when the dimension of the system is high. This paper discusses a method to obtain a reduced second-order model from a large sparse second-order index-3 system using the Balanced Truncation. For this purpose, the low-rank alternating direction implicit iteration is modified to solve the Lyapunov equations of the index-3 structure system efficiently in an implicit way. Numerical resultants are discussed to show the reflectivity and efficiency of the techniques.
- Nearly conservative multivalue methods with extended bounded parasitism
- Abstract: Publication date: Available online 12 December 2019Source: Applied Numerical MathematicsAuthor(s): Vincenzo Citro, Raffaele D'AmbrosioAbstractThe paper is focused on the analysis of parasitism for multivalue numerical methods intended as geometric numerical integrators for Hamiltonian problems. In particular, the main topic is the design of multivalue numerical methods whose parasitic components remain bounded over certain time intervals, opening the path to the development of nearly conservative multivalue methods able to guarantee a control of parasitism in the long time. The analysis of parasitism as well as the development of the corresponding methods is the core of the treatise. The effectiveness of the approach is also confirmed on selected Hamiltonian problems.
- Abstract: Publication date: Available online 12 December 2019Source: Applied Numerical MathematicsAuthor(s): Vincenzo Citro, Raffaele D'AmbrosioAbstractThe paper is focused on the analysis of parasitism for multivalue numerical methods intended as geometric numerical integrators for Hamiltonian problems. In particular, the main topic is the design of multivalue numerical methods whose parasitic components remain bounded over certain time intervals, opening the path to the development of nearly conservative multivalue methods able to guarantee a control of parasitism in the long time. The analysis of parasitism as well as the development of the corresponding methods is the core of the treatise. The effectiveness of the approach is also confirmed on selected Hamiltonian problems.
- Convergence analysis of a highly accurate Nyström scheme for Fredholm
integral equations- Abstract: Publication date: Available online 10 December 2019Source: Applied Numerical MathematicsAuthor(s): Fadi Awawdeh, Linda SmailAbstractA stable and convergent Nyström scheme is proposed to solve Fredholm integral equations (FIEs). Our approximation is based on the barycentric rational interpolants. By introducing barycentric quadratures to the integral operator that appears in the FIE and modifying the standard Nyström scheme, we demonstrate that the new Nyström scheme is a viable option for the numerical solution of FIEs. Convergence rates of the method are proved taking into account the effect of grading the domain. The final convergence result shows clearly that one can choose an optimal domain grading. Numerical examples and comparisons with competitive methods of tunable accuracy are provided to support the theoretical analysis and illustrate the efficiency of the proposed numerical scheme.
- Abstract: Publication date: Available online 10 December 2019Source: Applied Numerical MathematicsAuthor(s): Fadi Awawdeh, Linda SmailAbstractA stable and convergent Nyström scheme is proposed to solve Fredholm integral equations (FIEs). Our approximation is based on the barycentric rational interpolants. By introducing barycentric quadratures to the integral operator that appears in the FIE and modifying the standard Nyström scheme, we demonstrate that the new Nyström scheme is a viable option for the numerical solution of FIEs. Convergence rates of the method are proved taking into account the effect of grading the domain. The final convergence result shows clearly that one can choose an optimal domain grading. Numerical examples and comparisons with competitive methods of tunable accuracy are provided to support the theoretical analysis and illustrate the efficiency of the proposed numerical scheme.
- Diagonalized Gegenbauer rational spectral methods for second- and
fourth-order problems on the whole line- Abstract: Publication date: Available online 9 December 2019Source: Applied Numerical MathematicsAuthor(s): Shan Li, Zhenyan Lai, Lusha Jin, Xuhong YuAbstractFully diagonalized Gegenbauer rational spectral methods for solving second- and fourth-order differential equations on the whole line are proposed and analyzed. Some Gegenbauer rational Sobolev orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Gegenbauer rational series. Optimal error estimates of the fully diagonalized Gegenbauer rational spectral method for second-order problem are obtained. Finally, some numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized methods.
- Abstract: Publication date: Available online 9 December 2019Source: Applied Numerical MathematicsAuthor(s): Shan Li, Zhenyan Lai, Lusha Jin, Xuhong YuAbstractFully diagonalized Gegenbauer rational spectral methods for solving second- and fourth-order differential equations on the whole line are proposed and analyzed. Some Gegenbauer rational Sobolev orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Gegenbauer rational series. Optimal error estimates of the fully diagonalized Gegenbauer rational spectral method for second-order problem are obtained. Finally, some numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized methods.
- The adaptive finite element method for the P-Laplace problem
- Abstract: Publication date: Available online 6 December 2019Source: Applied Numerical MathematicsAuthor(s): D.J. Liu, Z.R. ChenAbstractWe consider the adaptive finite element method (AFEM) for the P-Laplace problem, −div( ∇u p−2∇u)=f. A posteriori and priori error analysis of conforming and nonconforming finite element method are measured in the new framework. A number of experiments confirm the effective decay rates of the AFEM.
- Abstract: Publication date: Available online 6 December 2019Source: Applied Numerical MathematicsAuthor(s): D.J. Liu, Z.R. ChenAbstractWe consider the adaptive finite element method (AFEM) for the P-Laplace problem, −div( ∇u p−2∇u)=f. A posteriori and priori error analysis of conforming and nonconforming finite element method are measured in the new framework. A number of experiments confirm the effective decay rates of the AFEM.
- A conjugate gradient algorithm and its applications in image restoration
- Abstract: Publication date: Available online 6 December 2019Source: Applied Numerical MathematicsAuthor(s): Junyue Cao, Jinzhao WuAbstractIn this paper, a nonlinear conjugate gradient algorithm is presented. This given algorithm possesses the following properties: (i) the sufficient descent property is satisfied; (ii) the trust region feature holds; (iii) the global convergence is proved for nonconvex functions; (iv) the applications for image restoration problem are done to show the performance of the proposed algorithm. Notice that the properties (i) and (ii) will be obtained without other special conditions.
- Abstract: Publication date: Available online 6 December 2019Source: Applied Numerical MathematicsAuthor(s): Junyue Cao, Jinzhao WuAbstractIn this paper, a nonlinear conjugate gradient algorithm is presented. This given algorithm possesses the following properties: (i) the sufficient descent property is satisfied; (ii) the trust region feature holds; (iii) the global convergence is proved for nonconvex functions; (iv) the applications for image restoration problem are done to show the performance of the proposed algorithm. Notice that the properties (i) and (ii) will be obtained without other special conditions.
- A HJB-POD approach for the control of nonlinear PDEs on a tree structure
- Abstract: Publication date: Available online 5 December 2019Source: Applied Numerical MathematicsAuthor(s): Alessandro Alla, Luca SaluzziAbstractThe Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.
- Abstract: Publication date: Available online 5 December 2019Source: Applied Numerical MathematicsAuthor(s): Alessandro Alla, Luca SaluzziAbstractThe Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.
- A Chebyshev pseudo-spectral method for the multi-dimensional fractional
Rayleigh problem for a generalized Maxwell fluid with Robin boundary
conditions- Abstract: Publication date: Available online 5 December 2019Source: Applied Numerical MathematicsAuthor(s): Shina D. Oloniiju, Sicelo P. Goqo, Precious SibandaAbstractMany pseudospectral schemes have been developed for time fractional partial differential equations, most of which use the spectral tau method. In this study, we develop an accurate numerical scheme using a combination of Lagrange and Chebyshev polynomials of the first kind for the multi-dimensional fractional Rayleigh problem integrated using the Gauss–Lobatto quadrature. The Chebyshev expansion coefficients are evaluated using the orthogonality condition of the polynomials. Arbitrary temporal derivatives are approximated using shifted Chebyshev polynomials, while the spatial derivatives are approximated using Lagrange basis functions. To establish the accuracy of the proposed scheme, we present a convergence analysis of the absolute errors. The convergence analysis shows that the absolute error tends to zero for sufficiently large number of collocation points.
- Abstract: Publication date: Available online 5 December 2019Source: Applied Numerical MathematicsAuthor(s): Shina D. Oloniiju, Sicelo P. Goqo, Precious SibandaAbstractMany pseudospectral schemes have been developed for time fractional partial differential equations, most of which use the spectral tau method. In this study, we develop an accurate numerical scheme using a combination of Lagrange and Chebyshev polynomials of the first kind for the multi-dimensional fractional Rayleigh problem integrated using the Gauss–Lobatto quadrature. The Chebyshev expansion coefficients are evaluated using the orthogonality condition of the polynomials. Arbitrary temporal derivatives are approximated using shifted Chebyshev polynomials, while the spatial derivatives are approximated using Lagrange basis functions. To establish the accuracy of the proposed scheme, we present a convergence analysis of the absolute errors. The convergence analysis shows that the absolute error tends to zero for sufficiently large number of collocation points.
- The revisited total least squares problems with linear equality constraint
- Abstract: Publication date: Available online 3 December 2019Source: Applied Numerical MathematicsAuthor(s): Qiaohua Liu, Shufang Jin, Lei Yao, Dongmei ShenAbstractThe total least squares problem with linear equality constraint is proved to be approximated by an unconstrained total least squares problem with a large weight on the constraint. A criterion for choosing the weighting factor is given, and a QR-based inverse (QR-INV) iteration method is presented. Numerical results show that the QR-INV method is more efficient than the standard QR-SVD procedure and Schaffrin's inverse iteration method, especially for large and sparse matrices.
- Abstract: Publication date: Available online 3 December 2019Source: Applied Numerical MathematicsAuthor(s): Qiaohua Liu, Shufang Jin, Lei Yao, Dongmei ShenAbstractThe total least squares problem with linear equality constraint is proved to be approximated by an unconstrained total least squares problem with a large weight on the constraint. A criterion for choosing the weighting factor is given, and a QR-based inverse (QR-INV) iteration method is presented. Numerical results show that the QR-INV method is more efficient than the standard QR-SVD procedure and Schaffrin's inverse iteration method, especially for large and sparse matrices.
- Discretization of fractional boundary value problems using split operator
local extension problems- Abstract: Publication date: Available online 30 November 2019Source: Applied Numerical MathematicsAuthor(s): John P. RoopAbstractPartial differential equations which include nonlocal operators have recently become a major focus of mathematical and computational research. The efficient computational implementation of a nonlocal operator remains an important question. In this article, we introduce an operator splitting approach to the discretization of nonlocal boundary value problems. It has been shown that the fractional Laplacian in Rd is identical to equivalent local extension problem in Rd+1 with a nonlinear Neumann boundary condition. We present an operator splitting approach to the extended problem that enables a consistent stable implementation. Computational experiments are presented using finite differences and meshless methods.
- Abstract: Publication date: Available online 30 November 2019Source: Applied Numerical MathematicsAuthor(s): John P. RoopAbstractPartial differential equations which include nonlocal operators have recently become a major focus of mathematical and computational research. The efficient computational implementation of a nonlocal operator remains an important question. In this article, we introduce an operator splitting approach to the discretization of nonlocal boundary value problems. It has been shown that the fractional Laplacian in Rd is identical to equivalent local extension problem in Rd+1 with a nonlinear Neumann boundary condition. We present an operator splitting approach to the extended problem that enables a consistent stable implementation. Computational experiments are presented using finite differences and meshless methods.
- The discontinuous Galerkin method for stochastic differential equations
driven by additive noises- Abstract: Publication date: Available online 29 November 2019Source: Applied Numerical MathematicsAuthor(s): Mahboub Baccouch, Helmi Temimi, Mohamed Ben-RomdhaneAbstractIn this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when p-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.
- Abstract: Publication date: Available online 29 November 2019Source: Applied Numerical MathematicsAuthor(s): Mahboub Baccouch, Helmi Temimi, Mohamed Ben-RomdhaneAbstractIn this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when p-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.
- A generalized-Jacobi-function spectral method for space-time fractional
reaction-diffusion equations with viscosity terms- Abstract: Publication date: Available online 28 November 2019Source: Applied Numerical MathematicsAuthor(s): Zhe Yu, Boying Wu, Jiebao Sun, Wenjie LiuAbstractIn this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.
- Abstract: Publication date: Available online 28 November 2019Source: Applied Numerical MathematicsAuthor(s): Zhe Yu, Boying Wu, Jiebao Sun, Wenjie LiuAbstractIn this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion.
- Soft computing technique for a system of fuzzy volterra
integro-differential equations in a Hilbert space- Abstract: Publication date: Available online 28 November 2019Source: Applied Numerical MathematicsAuthor(s): Ghaleb Gumah, Shrideh Al-Omari, Dumitru BaleanuAbstractIn this article, we implement a relatively new computational technique, a reproducing kernel Hilbert space method, for solving a system of fuzzy Volterra integro-differential equations in the Hilbert Space ⊕j=1n(W22[a,b]⊕W22[a,b]). Based on the concept of the reproducing kernel function combined with Gram-Schmidt orthogonalization process, we represent an exact solution in a form of Fourier series in the reproducing kernel Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Accordingly, the approximate solution of the system of fuzzy Volterra integro-differential equations is obtained by the n-term intercept of the exact solution and proved to converge to the exact solution. Finally, two numerical examples are presented to illustrate the reliability, appropriateness and efficiency of the method.
- Abstract: Publication date: Available online 28 November 2019Source: Applied Numerical MathematicsAuthor(s): Ghaleb Gumah, Shrideh Al-Omari, Dumitru BaleanuAbstractIn this article, we implement a relatively new computational technique, a reproducing kernel Hilbert space method, for solving a system of fuzzy Volterra integro-differential equations in the Hilbert Space ⊕j=1n(W22[a,b]⊕W22[a,b]). Based on the concept of the reproducing kernel function combined with Gram-Schmidt orthogonalization process, we represent an exact solution in a form of Fourier series in the reproducing kernel Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Accordingly, the approximate solution of the system of fuzzy Volterra integro-differential equations is obtained by the n-term intercept of the exact solution and proved to converge to the exact solution. Finally, two numerical examples are presented to illustrate the reliability, appropriateness and efficiency of the method.
- A fast linearized finite difference method for the nonlinear multi-term
time-fractional wave equation- Abstract: Publication date: Available online 27 November 2019Source: Applied Numerical MathematicsAuthor(s): Pin Lyu, Yuxiang Liang, Zhibo WangAbstractIn this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. The proposed fast linearized method can be directly extended to solve the nonlinear distributed-order time-fractional wave problem. Numerical examples are provided to justify the efficiency.
- Abstract: Publication date: Available online 27 November 2019Source: Applied Numerical MathematicsAuthor(s): Pin Lyu, Yuxiang Liang, Zhibo WangAbstractIn this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L2-1σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H1-norm. The proposed fast linearized method can be directly extended to solve the nonlinear distributed-order time-fractional wave problem. Numerical examples are provided to justify the efficiency.
- Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and
diffusion coefficients- Abstract: Publication date: Available online 26 November 2019Source: Applied Numerical MathematicsAuthor(s): Siqing Gan, Youzi He, Xiaojie WangAbstractTraditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly growing coefficients. Motivated by this, various modified versions of explicit Euler and Milstein methods were constructed and analyzed in the literature. In the present paper, we aim to introduce a family of explicit tamed stochastic Runge-Kutta (TSRK) methods for commutative SDEs with super-linearly growing drift and diffusion coefficients. Strong convergence rates of order 1.0 are successfully identified for the proposed methods under certain non-globally Lipschitz conditions. Compared to the Milstein-type methods involved with derivatives of coefficients, the newly proposed derivative-free TSRK methods can be computationally more efficient. Numerical experiments are reported to confirm the expected strong convergence rate of the TSRK methods.
- Abstract: Publication date: Available online 26 November 2019Source: Applied Numerical MathematicsAuthor(s): Siqing Gan, Youzi He, Xiaojie WangAbstractTraditional explicit schemes such as the Euler-Maruyama, Milstein and stochastic Runge-Kutta methods, in general, result in strong and weak divergence when solving stochastic differential equations (SDEs) with super-linearly growing coefficients. Motivated by this, various modified versions of explicit Euler and Milstein methods were constructed and analyzed in the literature. In the present paper, we aim to introduce a family of explicit tamed stochastic Runge-Kutta (TSRK) methods for commutative SDEs with super-linearly growing drift and diffusion coefficients. Strong convergence rates of order 1.0 are successfully identified for the proposed methods under certain non-globally Lipschitz conditions. Compared to the Milstein-type methods involved with derivatives of coefficients, the newly proposed derivative-free TSRK methods can be computationally more efficient. Numerical experiments are reported to confirm the expected strong convergence rate of the TSRK methods.
- Stability and convergence based on the finite difference method for the
nonlinear fractional cable equation on non-uniform staggered grids- Abstract: Publication date: Available online 25 November 2019Source: Applied Numerical MathematicsAuthor(s): Xiaoli Li, Hongxing RuiAbstractIn this article, a block-centered finite difference method for the nonlinear fractional cable equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δtα+h2+k2) both for pressure and velocity are established on non-uniform rectangular grids, where α=min{1+γ1,1+γ2}, Δt,h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
- Abstract: Publication date: Available online 25 November 2019Source: Applied Numerical MathematicsAuthor(s): Xiaoli Li, Hongxing RuiAbstractIn this article, a block-centered finite difference method for the nonlinear fractional cable equation is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O(Δtα+h2+k2) both for pressure and velocity are established on non-uniform rectangular grids, where α=min{1+γ1,1+γ2}, Δt,h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
- Space-time finite element method for the distributed-order time fractional
reaction diffusion equations- Abstract: Publication date: Available online 23 November 2019Source: Applied Numerical MathematicsAuthor(s): Weiping Bu, Lun Ji, Yifa Tang, Jie ZhouAbstractIn this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t=0, the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t−α,α∈(0,1) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method.
- Abstract: Publication date: Available online 23 November 2019Source: Applied Numerical MathematicsAuthor(s): Weiping Bu, Lun Ji, Yifa Tang, Jie ZhouAbstractIn this paper, we propose a space-time finite element method for the distributed-order time fractional reaction diffusion equations (DOTFRDEs). First, using the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature rule to discretize the distributed-order derivative and employing the finite element method both in space and time, three fully discrete finite element schemes for DOTFRDEs are developed. Second, for the obtained numerical schemes, the existence, uniqueness and stability are discussed. Third, under the hypothesis about the singular behavior of exact solution near t=0, the convergence of these numerical schemes are investigated in detail based on the graded time mesh. Forth, in order to reduce the storage requirement and computational cost of these numerical schemes, an efficient sum-of-exponentials approximation for the kernel t−α,α∈(0,1) is introduced, and the developed space-time finite element schemes are improved. At last, some numerical tests are given to verify the rationality and effectiveness of our method.
- Efficient iterative solvers for a complex valued two-by-two block linear
system with application to parabolic optimal control problems- Abstract: Publication date: Available online 22 November 2019Source: Applied Numerical MathematicsAuthor(s): Zhao-Zheng Liang, Owe Axelsson, Guo-Feng ZhangAbstractIn this paper, we are concerned with the efficient iterative solution of time-harmonic parabolic optimal control problems. A robust parameterized preconditioner is proposed for the arising complex valued two-by-two block linear system related to the first-order optimality conditions. Practical parameter choice strategies are considered for the new preconditioner to improve the performance of the original preconditioner within Krylov subspace acceleration. Moreover, a nonstationary second-order iteration method is devised from the parameterized preconditioner within Chebyshev acceleration. Based on a detailed spectral analysis of the preconditioned matrix, convergence rates are analyzed for both the established Krylov subspace and Chebyshev acceleration methods. Due to the tight and problem independent eigenvalue distributions of the preconditioned matrix, the implementation of the Chebyshev acceleration method is parameter free and the obtained iteration error bounds of both methods result in almost parameter independent convergence rates. Numerical experiments are presented to confirm the robustness and effectiveness of the parameterized preconditioner for both Krylov subspace and Chebyshev accelerations and improvement compared to earlier results.
- Abstract: Publication date: Available online 22 November 2019Source: Applied Numerical MathematicsAuthor(s): Zhao-Zheng Liang, Owe Axelsson, Guo-Feng ZhangAbstractIn this paper, we are concerned with the efficient iterative solution of time-harmonic parabolic optimal control problems. A robust parameterized preconditioner is proposed for the arising complex valued two-by-two block linear system related to the first-order optimality conditions. Practical parameter choice strategies are considered for the new preconditioner to improve the performance of the original preconditioner within Krylov subspace acceleration. Moreover, a nonstationary second-order iteration method is devised from the parameterized preconditioner within Chebyshev acceleration. Based on a detailed spectral analysis of the preconditioned matrix, convergence rates are analyzed for both the established Krylov subspace and Chebyshev acceleration methods. Due to the tight and problem independent eigenvalue distributions of the preconditioned matrix, the implementation of the Chebyshev acceleration method is parameter free and the obtained iteration error bounds of both methods result in almost parameter independent convergence rates. Numerical experiments are presented to confirm the robustness and effectiveness of the parameterized preconditioner for both Krylov subspace and Chebyshev accelerations and improvement compared to earlier results.
- Energy preserving relaxation method for space-fractional nonlinear
Schrödinger equation- Abstract: Publication date: Available online 21 November 2019Source: Applied Numerical MathematicsAuthor(s): Bianru Cheng, Dongling Wang, Wei YangAbstractIn this paper, we develop and analyze an energy preserving relaxation scheme for a class of space fractional nonlinear Schrödinger equations (F-NLS) with periodic boundary condition. This scheme can preserve the discrete mass and energy for F-NLS with general power nonlinearity. The second order convergence in time direction for the numerical method is rigorously proved for the cubic nonlinear F-NLS. Some numerical simulations for 1D and 2D cases based on the Fourier spectral approximation in space are presented to validate the theoretical analysis of the proposed method.
- Abstract: Publication date: Available online 21 November 2019Source: Applied Numerical MathematicsAuthor(s): Bianru Cheng, Dongling Wang, Wei YangAbstractIn this paper, we develop and analyze an energy preserving relaxation scheme for a class of space fractional nonlinear Schrödinger equations (F-NLS) with periodic boundary condition. This scheme can preserve the discrete mass and energy for F-NLS with general power nonlinearity. The second order convergence in time direction for the numerical method is rigorously proved for the cubic nonlinear F-NLS. Some numerical simulations for 1D and 2D cases based on the Fourier spectral approximation in space are presented to validate the theoretical analysis of the proposed method.
- The local discontinuous Galerkin finite element methods for Caputo-type
partial differential equations: Mathematical analysis- Abstract: Publication date: Available online 16 November 2019Source: Applied Numerical MathematicsAuthor(s): Changpin Li, Zhen WangAbstractIn [Li and Wang, Appl. Numer. Math. 140 (2019) 1-22], three kinds of typical Caputo-type partial differential equations are numerically studied via the local discontinuous Galerkin (LDG) finite element methods, including Caputo-type reaction-diffusion equation, Caputo-type reaction-diffusion-wave equation, and Caputo-type cable equation without regularity analysis. In this article, we study the existence, uniqueness, and regularity of the solutions to these equations. Besides, the error estimate for Caputo-type cable equation has been greatly improved.
- Abstract: Publication date: Available online 16 November 2019Source: Applied Numerical MathematicsAuthor(s): Changpin Li, Zhen WangAbstractIn [Li and Wang, Appl. Numer. Math. 140 (2019) 1-22], three kinds of typical Caputo-type partial differential equations are numerically studied via the local discontinuous Galerkin (LDG) finite element methods, including Caputo-type reaction-diffusion equation, Caputo-type reaction-diffusion-wave equation, and Caputo-type cable equation without regularity analysis. In this article, we study the existence, uniqueness, and regularity of the solutions to these equations. Besides, the error estimate for Caputo-type cable equation has been greatly improved.
- Global inexact quasi-Newton method for nonlinear system of equations with
constraints- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): C.A. Arias, H.J. Martínez, R. PérezAbstractIn this paper, we propose a Projected Inexact Quasi-Newton Method (PIQN) to solve nonlinear systems of equations with nonnegative constraints and we develop its respective global convergence theory. Particularly, we show that the method enjoys superlinear convergence rate. In addition, we show the good numerical performance of the method when applied to problems that arise in chemistry, engineering and mathematics. We highlight the efficiency of the method to solve Nonlinear Complementarity Problems (NCP), which appear in many application fields and are recognized as hard problems.
- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): C.A. Arias, H.J. Martínez, R. PérezAbstractIn this paper, we propose a Projected Inexact Quasi-Newton Method (PIQN) to solve nonlinear systems of equations with nonnegative constraints and we develop its respective global convergence theory. Particularly, we show that the method enjoys superlinear convergence rate. In addition, we show the good numerical performance of the method when applied to problems that arise in chemistry, engineering and mathematics. We highlight the efficiency of the method to solve Nonlinear Complementarity Problems (NCP), which appear in many application fields and are recognized as hard problems.
- Strong stability preserving second derivative diagonally implicit
multistage integration methods- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): A. Moradi, A. Abdi, J. FarziAbstractConstruction of strong stability preserving second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) with and without quadratic stability (QS) has been investigated. Examples of such methods with p=q=r=s=2,3 and 4 have been constructed, where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. To show the efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in preserving some nonlinear stability properties such as total variation and positivity, several one dimensional numerical experiments are performed.
- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): A. Moradi, A. Abdi, J. FarziAbstractConstruction of strong stability preserving second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) with and without quadratic stability (QS) has been investigated. Examples of such methods with p=q=r=s=2,3 and 4 have been constructed, where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. To show the efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in preserving some nonlinear stability properties such as total variation and positivity, several one dimensional numerical experiments are performed.
- On subgrid multiscale stabilized finite element method for
advection-diffusion-reaction equation with variable coefficients- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): Manisha Chowdhury, B.V. Rathish KumarAbstractIn this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been introduced. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen to stabilize the Galerkin finite element method. Both a priori and a posteriori finite element error estimates in L2 norm have been derived after introducing the stabilized variational form. An expression of the stabilization parameter has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and the credibility of the theoretically derived expression of the stabilization parameter has been established numerically.
- Abstract: Publication date: Available online 6 November 2019Source: Applied Numerical MathematicsAuthor(s): Manisha Chowdhury, B.V. Rathish KumarAbstractIn this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been introduced. Here subgrid scale approach along with algebraic approximation to the sub-scales has been chosen to stabilize the Galerkin finite element method. Both a priori and a posteriori finite element error estimates in L2 norm have been derived after introducing the stabilized variational form. An expression of the stabilization parameter has also been derived here. At last numerical experiments are presented to verify numerical performance of the stabilized method and the credibility of the theoretically derived expression of the stabilization parameter has been established numerically.
- Two fast and efficient linear semi-implicit approaches with unconditional
energy stability for nonlocal phase field crystal equation.- Abstract: Publication date: Available online 31 October 2019Source: Applied Numerical MathematicsAuthor(s): Zhengguang Liu, Xiaoli LiAbstractThe phase-field crystal equation is a sixth-order nonlinear parabolic equation and have received increasing attention in the study of the microstructural evolution of two-phase systems on atomic length and diffusive time scales. This model can be applied to simulate various phenomena such as epitaxial growth, material hardness and phase transition. Compared with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal phase-field crystal equation equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. We propose linear semi-implicit approach and scalar auxiliary variable approach with unconditional energy stability for the nonlocal phase-field crystal equation. The first contribution is that we have proved the unconditional energy stability for nonlocal phase-field crystal model and its semi-discrete schemes carefully and rigorously. Secondly, we found a fast procedure to reduce the computational work and memory requirement which the non-locality of the nonlocal diffusion term generates huge computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.
- Abstract: Publication date: Available online 31 October 2019Source: Applied Numerical MathematicsAuthor(s): Zhengguang Liu, Xiaoli LiAbstractThe phase-field crystal equation is a sixth-order nonlinear parabolic equation and have received increasing attention in the study of the microstructural evolution of two-phase systems on atomic length and diffusive time scales. This model can be applied to simulate various phenomena such as epitaxial growth, material hardness and phase transition. Compared with the classical local gradient flow and phase field models, the nonlocal models such as nonlocal phase-field crystal equation equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions. We propose linear semi-implicit approach and scalar auxiliary variable approach with unconditional energy stability for the nonlocal phase-field crystal equation. The first contribution is that we have proved the unconditional energy stability for nonlocal phase-field crystal model and its semi-discrete schemes carefully and rigorously. Secondly, we found a fast procedure to reduce the computational work and memory requirement which the non-locality of the nonlocal diffusion term generates huge computational work and memory requirement. Finally, several numerical simulations are demonstrated to verify the accuracy and efficiency of our proposed schemes.
- Numerical solution of nonlinear 2D optimal control problems generated by
Atangana-Riemann-Liouville fractal-fractional derivative- Abstract: Publication date: Available online 31 October 2019Source: Applied Numerical MathematicsAuthor(s): M.H. HeydariAbstractThis paper is dedicated to introduce a new category of nonlinear 2D optimal control problems (OCPs), generated by dynamical systems involved with fractal-fractional derivatives and to develop an operational matrix (OM) scheme for their numerical solutions. The fractal-fractional derivatives are defined in the Atangana-Riemann-Liouville sense with Mittag-Leffler non-singular kernel. The proposed approach is based on the Chelyshkov polynomials (CPs) and their OM of fractal-fractional derivative (which is derived in the present study). More precisely, the proposed method transforms solving such problems into solving systems of nonlinear algebraic equations. To this end, at first the state and control variables are approximated by the CPs with unknown coefficients, and are substituted in the objective function, dynamical system and the initial conditions. Then, the 2D Gauss-Legendre quadrature rule with the OM of fractal-fractional derivative of the CPs are utilized to construct a constrained problem, which is solved by the Lagrange multipliers method. The accuracy of the presented method is investigated on two test problems. The obtained results confirm that the presented scheme is highly accurate for the numerical solution of such problems.
- Abstract: Publication date: Available online 31 October 2019Source: Applied Numerical MathematicsAuthor(s): M.H. HeydariAbstractThis paper is dedicated to introduce a new category of nonlinear 2D optimal control problems (OCPs), generated by dynamical systems involved with fractal-fractional derivatives and to develop an operational matrix (OM) scheme for their numerical solutions. The fractal-fractional derivatives are defined in the Atangana-Riemann-Liouville sense with Mittag-Leffler non-singular kernel. The proposed approach is based on the Chelyshkov polynomials (CPs) and their OM of fractal-fractional derivative (which is derived in the present study). More precisely, the proposed method transforms solving such problems into solving systems of nonlinear algebraic equations. To this end, at first the state and control variables are approximated by the CPs with unknown coefficients, and are substituted in the objective function, dynamical system and the initial conditions. Then, the 2D Gauss-Legendre quadrature rule with the OM of fractal-fractional derivative of the CPs are utilized to construct a constrained problem, which is solved by the Lagrange multipliers method. The accuracy of the presented method is investigated on two test problems. The obtained results confirm that the presented scheme is highly accurate for the numerical solution of such problems.
- Some three-term conjugate gradient methods with the new direction
structure- Abstract: Publication date: Available online 30 October 2019Source: Applied Numerical MathematicsAuthor(s): J.K. Liu, Y.X. Zhao, X.L. WuAbstractIn this paper, a three-term conjugate gradient method with the new direction structure is proposed for solving large-scale unconstrained optimization problems, which generates a sufficient descent direction in per-iteration by the aid of some inexact line search conditions. Under suitable assumptions, the proposed method is globally convergent for nonconvex smooth problems. We further generalize the new direction structure to other traditional methods and obtain some algorithms with the same structure as the proposed method. Preliminary numerical comparisons show that the proposed methods are effective and promising for the test problems.
- Abstract: Publication date: Available online 30 October 2019Source: Applied Numerical MathematicsAuthor(s): J.K. Liu, Y.X. Zhao, X.L. WuAbstractIn this paper, a three-term conjugate gradient method with the new direction structure is proposed for solving large-scale unconstrained optimization problems, which generates a sufficient descent direction in per-iteration by the aid of some inexact line search conditions. Under suitable assumptions, the proposed method is globally convergent for nonconvex smooth problems. We further generalize the new direction structure to other traditional methods and obtain some algorithms with the same structure as the proposed method. Preliminary numerical comparisons show that the proposed methods are effective and promising for the test problems.
- On accurate solution of the Fredholm integral equations of the second kind
- Abstract: Publication date: Available online 28 October 2019Source: Applied Numerical MathematicsAuthor(s): Sadegh Amiri, Mojtaba Hajipour, Dumitru BaleanuAbstractIn this paper, an accurate numerical method based on the cosine-trigonometric basis functions is developed to solve the Fredholm integral equations of the second kind. By using the proposed method, the presented equation is converted into a system of algebraic equations. The convergence analysis of the proposed method is also investigated. To demonstrate the efficiency of the proposed method, the numerical simulations of various types of one- and two-dimensional examples are prepared. Comparative results show that this method is accurate than the other existing methods in the literature.
- Abstract: Publication date: Available online 28 October 2019Source: Applied Numerical MathematicsAuthor(s): Sadegh Amiri, Mojtaba Hajipour, Dumitru BaleanuAbstractIn this paper, an accurate numerical method based on the cosine-trigonometric basis functions is developed to solve the Fredholm integral equations of the second kind. By using the proposed method, the presented equation is converted into a system of algebraic equations. The convergence analysis of the proposed method is also investigated. To demonstrate the efficiency of the proposed method, the numerical simulations of various types of one- and two-dimensional examples are prepared. Comparative results show that this method is accurate than the other existing methods in the literature.
- Inverse matrices for pseudospectral differentiation operators in polar
coordinates by stepwise integrations and low-rank updates- Abstract: Publication date: Available online 25 October 2019Source: Applied Numerical MathematicsAuthor(s): Yung-Ta Li, Po-Yu Lin, Chun-Hao TengAbstractSpectral/pseudospectral integration preconditioning matrices are useful tools for solving differential equations involving pure differential operators dm/dxm. In this study we construct well-conditioned inverse pseudospectral matrices for the basic differential operator and the mixed differential operator ddra(r)ddr on Gauss-Radau-Legendre points based on stepwise integrations and low-rank updates. The inverse matrices can be used either as a solution operator or an effective preconditioner for variable coefficient differential equations of first and second order in polar coordinates. Numerical experiments were conducted and we observed the performance of the inverse operator as expected.
- Abstract: Publication date: Available online 25 October 2019Source: Applied Numerical MathematicsAuthor(s): Yung-Ta Li, Po-Yu Lin, Chun-Hao TengAbstractSpectral/pseudospectral integration preconditioning matrices are useful tools for solving differential equations involving pure differential operators dm/dxm. In this study we construct well-conditioned inverse pseudospectral matrices for the basic differential operator and the mixed differential operator ddra(r)ddr on Gauss-Radau-Legendre points based on stepwise integrations and low-rank updates. The inverse matrices can be used either as a solution operator or an effective preconditioner for variable coefficient differential equations of first and second order in polar coordinates. Numerical experiments were conducted and we observed the performance of the inverse operator as expected.
- Galerkin spectral approximation of optimal control problems with L
2-norm
control constraint- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Xiuxiu Lin, Yanping Chen, Yunqing HuangAbstractIn this paper, an optimal control problem governed by elliptic partial differential equations, whose objective functionals do not depend on the controls, with L2-norm constraint on control variable is considered, and Galerkin spectral approximation of the optimal control problem without penalty term is established. To analyze the optimal control problem, optimality conditions of the control problem are first derived in detail. By virtue of some lemmas and auxiliary equations, a priori error estimates of spectral approximation for optimal control problems are analyzed rigorously. Moreover, a posteriori error estimates of the optimal control problem are also established carefully. In the end, some numerical examples are carried out to confirm the analytical results that the errors decay exponentially fast as long as the data is smooth.
- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Xiuxiu Lin, Yanping Chen, Yunqing HuangAbstractIn this paper, an optimal control problem governed by elliptic partial differential equations, whose objective functionals do not depend on the controls, with L2-norm constraint on control variable is considered, and Galerkin spectral approximation of the optimal control problem without penalty term is established. To analyze the optimal control problem, optimality conditions of the control problem are first derived in detail. By virtue of some lemmas and auxiliary equations, a priori error estimates of spectral approximation for optimal control problems are analyzed rigorously. Moreover, a posteriori error estimates of the optimal control problem are also established carefully. In the end, some numerical examples are carried out to confirm the analytical results that the errors decay exponentially fast as long as the data is smooth.
- Superconvergence of the gradient approximation for weak Galerkin finite
element methods on nonuniform rectangular partitions- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Dan Li, Chunmei Wang, Junping WangAbstractThis article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O(hr), 1.5≤r≤2, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O(h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.
- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Dan Li, Chunmei Wang, Junping WangAbstractThis article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O(hr), 1.5≤r≤2, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O(h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.
- Error analysis of a time-splitting method for incompressible flows with
variable density- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Rong AnAbstractThis work is devoted to obtain optimal temporal error estimates for a first-order time-splitting scheme for solving the incompressible Navier-Stokes equations with variable density. This scheme was firstly proposed by Li, Mei et al. (Li, Mei et al., Journal of Computational Physics, 242(2013) 124-137). First-order convergence rates have been numerically observed, but never hitherto proved. In this work, we will give a rigorous temporal error analysis for this scheme under some regularity assumptions.
- Abstract: Publication date: Available online 23 October 2019Source: Applied Numerical MathematicsAuthor(s): Rong AnAbstractThis work is devoted to obtain optimal temporal error estimates for a first-order time-splitting scheme for solving the incompressible Navier-Stokes equations with variable density. This scheme was firstly proposed by Li, Mei et al. (Li, Mei et al., Journal of Computational Physics, 242(2013) 124-137). First-order convergence rates have been numerically observed, but never hitherto proved. In this work, we will give a rigorous temporal error analysis for this scheme under some regularity assumptions.
- A combined scheme of the local spectral element method and the generalized
plane wave discontinuous Galerkin method for the anisotropic Helmholtz
equation- Abstract: Publication date: Available online 22 October 2019Source: Applied Numerical MathematicsAuthor(s): Long YuanAbstractIn this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the approximation solutions generated by the proposed discretization method are derived. Especially, the orders of the condition number ρ of the anisotropic matrix in the error estimates are optimal. Finally, numerical results verify the validity of the theoretical results, and indicate that the new method possesses high accuracy and is slightly affected by the pollution effect for the large wavenumbers.
- Abstract: Publication date: Available online 22 October 2019Source: Applied Numerical MathematicsAuthor(s): Long YuanAbstractIn this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the approximation solutions generated by the proposed discretization method are derived. Especially, the orders of the condition number ρ of the anisotropic matrix in the error estimates are optimal. Finally, numerical results verify the validity of the theoretical results, and indicate that the new method possesses high accuracy and is slightly affected by the pollution effect for the large wavenumbers.