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Computers & Mathematics with Applications
Journal Prestige (SJR): 1.058 Citation Impact (citeScore): 2 Number of Followers: 11 Subscription journal ISSN (Print) 08981221  ISSN (Online) 08981221 Published by Elsevier [3206 journals] 
 On a convergent DSA preconditioned source iteration for a DGFEM method for
radiative transfer Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Olena Palii, Matthias SchlottbomAbstractWe consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The evenparity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinitedimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for planeparallel geometries with isotropic scattering, but the approach and proofs generalize to multidimensional problems and more general scattering operators verbatim.
 Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Olena Palii, Matthias SchlottbomAbstractWe consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The evenparity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinitedimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for planeparallel geometries with isotropic scattering, but the approach and proofs generalize to multidimensional problems and more general scattering operators verbatim.
 Opensource immersogeometric analysis of fluid–structure interaction
using FEniCS and tIGAr Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): David KamenskyAbstractWe recently developed the opensource library tIGAr, which extends the FEniCS finite element automation framework to isogeometric analysis. The present contribution demonstrates the utility of tIGAr in complex problems by applying it to immersogeometric fluid–structure interaction (FSI) analysis. This application is implemented as the new opensource library CouDALFISh (Coupling, via Dynamic Augmented Lagrangian, of Fluids with Immersed Shells, pronounced “cuttlefish”), which uses the dynamic augmented Lagrangian (DAL) method to couple fluid and shell structure subproblems. The DAL method was introduced previously, over a series of papers largely focused on heart valve FSI, but an opensource implementation making extensive use of automation to compile numerical routines from highlevel mathematical descriptions brings newfound transparency and reproducibility to these earlier developments on immersogeometric FSI analysis. The portions of CouDALFISh that do not use code generation also illustrate how a framework like FEniCS remains useful even when some functionality is outside the scope of its standard workflow. This paper summarizes the workings of CouDALFISh and documents a variety of benchmarks demonstrating its accuracy. Although the implementation emphasizes transparency and extensibility over performance, it is nonetheless demonstrated to be sufficient to simulate 3D FSI of an idealized aortic heart valve. Source code will be maintained at https://github.com/davidkamensky/CouDALFISh.
 Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): David KamenskyAbstractWe recently developed the opensource library tIGAr, which extends the FEniCS finite element automation framework to isogeometric analysis. The present contribution demonstrates the utility of tIGAr in complex problems by applying it to immersogeometric fluid–structure interaction (FSI) analysis. This application is implemented as the new opensource library CouDALFISh (Coupling, via Dynamic Augmented Lagrangian, of Fluids with Immersed Shells, pronounced “cuttlefish”), which uses the dynamic augmented Lagrangian (DAL) method to couple fluid and shell structure subproblems. The DAL method was introduced previously, over a series of papers largely focused on heart valve FSI, but an opensource implementation making extensive use of automation to compile numerical routines from highlevel mathematical descriptions brings newfound transparency and reproducibility to these earlier developments on immersogeometric FSI analysis. The portions of CouDALFISh that do not use code generation also illustrate how a framework like FEniCS remains useful even when some functionality is outside the scope of its standard workflow. This paper summarizes the workings of CouDALFISh and documents a variety of benchmarks demonstrating its accuracy. Although the implementation emphasizes transparency and extensibility over performance, it is nonetheless demonstrated to be sufficient to simulate 3D FSI of an idealized aortic heart valve. Source code will be maintained at https://github.com/davidkamensky/CouDALFISh.
 Isogeometric analysis for geometric modelling and acoustic attenuation
performances of reactive mufflers Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yaqiang Xue, Guoyong Jin, Tiangui Ye, Kangkang Shi, Saifeng Zhong, Chuanmeng YangAbstractIn the present study, the acoustic attenuation performances of mufflers are predicted by employing threedimensional isogeometric analysis (IGA) and multipatch technique for the first time. Circular, elliptical and conical reactive expansion chamber mufflers are exactly parameterized by nonuniform rational Bsplines (NURBS), and the Helmholtz equation governing the interior acoustic field is also solved by NURBS. Compared with traditional finite element method (FEM) which is rooted in Lagrange and Hermite functions, IGA can effectively avoid the timeconsuming meshing and totally preserve exact geometrical information during the modellinganalysis process. The computational effectiveness of multipatch IGA for calculating transmission loss is verified by comparing with experimental approach, boundary element method and other numerical methods from available research articles. The effects of geometry, such as shapes of the expansion cavity and lengths of the extended inlet/outlet are discussed in detail.
 Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yaqiang Xue, Guoyong Jin, Tiangui Ye, Kangkang Shi, Saifeng Zhong, Chuanmeng YangAbstractIn the present study, the acoustic attenuation performances of mufflers are predicted by employing threedimensional isogeometric analysis (IGA) and multipatch technique for the first time. Circular, elliptical and conical reactive expansion chamber mufflers are exactly parameterized by nonuniform rational Bsplines (NURBS), and the Helmholtz equation governing the interior acoustic field is also solved by NURBS. Compared with traditional finite element method (FEM) which is rooted in Lagrange and Hermite functions, IGA can effectively avoid the timeconsuming meshing and totally preserve exact geometrical information during the modellinganalysis process. The computational effectiveness of multipatch IGA for calculating transmission loss is verified by comparing with experimental approach, boundary element method and other numerical methods from available research articles. The effects of geometry, such as shapes of the expansion cavity and lengths of the extended inlet/outlet are discussed in detail.
 A matrixfree Chimera approach based on Dirichlet–Dirichlet coupling for
domain composition purposes Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Bruno Storti, Luciano Garelli, Mario Storti, Jorge D’ElíaAbstractA novel approach using an algebraicbased Chimera type domain composition method in the context of the finite element method for nonmatching overlapping unstructured grids is proposed in this work. The scheme is based on both the transfer of information across each grid interface via Dirichlet boundary conditions and a highorder interpolation algorithm to obtain one global solution of the system. The solution can be obtained iteratively, with a convergence rate that is similar to that obtained with an analogous conformal mesh, and the matrix–vector operator can be computed with completely decoupled operations on both meshes. Furthermore, the scheme can be set as a linear operator that can be fed to a matrixfree efficient iterative solver, such as the BiConjugate Gradient Stabilized method. Several numerical examples using nonmatching unstructured grids that are partially and completely overlapped with different element sizes are presented, assessing the precision and convergence rate of the method.
 Abstract: Publication date: Available online 22 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Bruno Storti, Luciano Garelli, Mario Storti, Jorge D’ElíaAbstractA novel approach using an algebraicbased Chimera type domain composition method in the context of the finite element method for nonmatching overlapping unstructured grids is proposed in this work. The scheme is based on both the transfer of information across each grid interface via Dirichlet boundary conditions and a highorder interpolation algorithm to obtain one global solution of the system. The solution can be obtained iteratively, with a convergence rate that is similar to that obtained with an analogous conformal mesh, and the matrix–vector operator can be computed with completely decoupled operations on both meshes. Furthermore, the scheme can be set as a linear operator that can be fed to a matrixfree efficient iterative solver, such as the BiConjugate Gradient Stabilized method. Several numerical examples using nonmatching unstructured grids that are partially and completely overlapped with different element sizes are presented, assessing the precision and convergence rate of the method.
 Design and investigation of cooling system for highpower LED luminaire
 Abstract: Publication date: Available online 20 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): K. Delendik, N. Kolyago, O. VoitikAbstractThis paper is concerned with development of cooling system for highpower LED luminaire, including design, simulation, production and investigation. Dimension, shape, and topology optimization algorithms and their program realization are developed with the use of MathCad. Thermal design by Comsol Multiphysics showed that the system on the base of heat pipes is the most effective system for cooling of highpower LED luminaire. The numerical simulation was employed to valid the designedin engineering solution of cooling system for highpower LED luminaire under elevated temperature operating conditions and different orientation in space. In accordance to simulation data cooling system on the base of heat pipes is designed. Theoretical results are well validated by experimental data and numerical simulation and can be widely utilized for designing of cooling system related to various LED products.
 Abstract: Publication date: Available online 20 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): K. Delendik, N. Kolyago, O. VoitikAbstractThis paper is concerned with development of cooling system for highpower LED luminaire, including design, simulation, production and investigation. Dimension, shape, and topology optimization algorithms and their program realization are developed with the use of MathCad. Thermal design by Comsol Multiphysics showed that the system on the base of heat pipes is the most effective system for cooling of highpower LED luminaire. The numerical simulation was employed to valid the designedin engineering solution of cooling system for highpower LED luminaire under elevated temperature operating conditions and different orientation in space. In accordance to simulation data cooling system on the base of heat pipes is designed. Theoretical results are well validated by experimental data and numerical simulation and can be widely utilized for designing of cooling system related to various LED products.
 Boundary integral multitrace formulations and Optimised Schwarz Methods
 Abstract: Publication date: Available online 19 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): X. Claeys, P. MarchandAbstractIn the present contribution, we consider Helmholtz equation with material coefficients being constant in each subdomain of a geometric partition of the propagation medium (discarding the presence of junctions), and we are interested in the numerical solution of such a problem by means of local multitrace boundary integral formulations (localMTF). For a one dimensional problem and configurations with two subdomains, it has been recently established that applying a Jacobi iterative solver to localMTF is exactly equivalent to an Optimised Schwarz Method (OSM) with a nonlocal impedance. In the present contribution, we show that this correspondence still holds in the case where the subdomain partition involves an arbitrary number of subdomains. From this, we deduce that the depth of the adjacency graph of the subdomain partition plays a critical role in the convergence of linear solvers applied to localMTF: we prove it for the case of homogeneous propagation medium and show, through numerical evidences, that this conclusion still holds for heterogeneous media. Our study also shows that, considering variants of localMTF involving a relaxation parameter, there is a fixed value of this relaxation parameter that systematically leads to optimal speed of convergence for linear solvers.
 Abstract: Publication date: Available online 19 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): X. Claeys, P. MarchandAbstractIn the present contribution, we consider Helmholtz equation with material coefficients being constant in each subdomain of a geometric partition of the propagation medium (discarding the presence of junctions), and we are interested in the numerical solution of such a problem by means of local multitrace boundary integral formulations (localMTF). For a one dimensional problem and configurations with two subdomains, it has been recently established that applying a Jacobi iterative solver to localMTF is exactly equivalent to an Optimised Schwarz Method (OSM) with a nonlocal impedance. In the present contribution, we show that this correspondence still holds in the case where the subdomain partition involves an arbitrary number of subdomains. From this, we deduce that the depth of the adjacency graph of the subdomain partition plays a critical role in the convergence of linear solvers applied to localMTF: we prove it for the case of homogeneous propagation medium and show, through numerical evidences, that this conclusion still holds for heterogeneous media. Our study also shows that, considering variants of localMTF involving a relaxation parameter, there is a fixed value of this relaxation parameter that systematically leads to optimal speed of convergence for linear solvers.
 Fractional sliding mode based on RBF neural network observer: Application
to HIV infection mathematical model Abstract: Publication date: Available online 18 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Amin Sharafian, Alireza Sharifi, Weidong ZhangAbstractA novel fractional observer based on sliding mode method combined with radial basis function neural network (RBFNN) is presented to distinguish the uncertainties of fractional order HIV (human immunodeficiency virus) mathematical dynamic model. Considering that the dynamical model of HIV infection varies from person to person (including the symptoms and the complex conditions), the mathematical model would be affected by uncertainties and disturbances. Therefore, in the proposed method, system uncertainties are estimated by the universal approximation ability of the neural network and external disturbance is vanquished by the robust attribute of sliding mode observer. The objectives are: firstly, to propose a method for estimating the states for nonlinear fractional order system based on a novel fractional update law for RBFNN. Secondly, by applying the fractional extension of Lyapunov stability called MittagLeffler stability, guarantee the finite time stability of the proposed observer. Finally, applying Matlab simulation, validate the performance of the proposed observer on HIV system.
 Abstract: Publication date: Available online 18 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Amin Sharafian, Alireza Sharifi, Weidong ZhangAbstractA novel fractional observer based on sliding mode method combined with radial basis function neural network (RBFNN) is presented to distinguish the uncertainties of fractional order HIV (human immunodeficiency virus) mathematical dynamic model. Considering that the dynamical model of HIV infection varies from person to person (including the symptoms and the complex conditions), the mathematical model would be affected by uncertainties and disturbances. Therefore, in the proposed method, system uncertainties are estimated by the universal approximation ability of the neural network and external disturbance is vanquished by the robust attribute of sliding mode observer. The objectives are: firstly, to propose a method for estimating the states for nonlinear fractional order system based on a novel fractional update law for RBFNN. Secondly, by applying the fractional extension of Lyapunov stability called MittagLeffler stability, guarantee the finite time stability of the proposed observer. Finally, applying Matlab simulation, validate the performance of the proposed observer on HIV system.
 A note on preconditioning for the 3 × 3 block saddle point
problem Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Xian Xie, HouBiao LiAbstractIn this paper, we mainly propose some new preconditioners for a special class of 3 × 3 block saddle point problems, which can arise from the timedependent Maxwell equations and some other practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumptions. Then, we analyze the corresponding eigenvalues of the new preconditioned matrices presented in this work. Furthermore, we show that proposed preconditioned matrices only have at most three distinct eigenvalues. Finally, two numerical examples are also carried to verify the effectiveness of proposed preconditioners.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Xian Xie, HouBiao LiAbstractIn this paper, we mainly propose some new preconditioners for a special class of 3 × 3 block saddle point problems, which can arise from the timedependent Maxwell equations and some other practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumptions. Then, we analyze the corresponding eigenvalues of the new preconditioned matrices presented in this work. Furthermore, we show that proposed preconditioned matrices only have at most three distinct eigenvalues. Finally, two numerical examples are also carried to verify the effectiveness of proposed preconditioners.
 Two meshless methods based on local radial basis function and barycentric
rational interpolation for solving 2D viscoelastic wave equation Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ömer OruçAbstractIn this paper, 2D viscoelastic wave equation is solved numerically both on regular and irregular domains. For spatial approximation of viscoelastic wave equation two meshless methods based on local radial basis function and barycentric rational interpolation are proposed. Both of the spatial approximation methods do not need mesh, node connectivity or integration process on local subdomains so they are truly meshless. For local radial basis function method we used an existing algorithm in literature to choose an acceptable shape parameter. Time marching is performed with fourth order Runge Kutta method. L2 and L∞ error norms for some test problems are reckoned to indicate efficiency and performance of the proposed two methods. Also, stability of the methods is discussed. Acquired results confirm the applicability of the proposed methods for 2D viscoelastic wave equation. We have performed some comparisons between the proposed two methods in the sense of accuracy and computational cost. From the comparisons, we have observed that performance of the barycentric rational interpolation in the sense of accuracy is slightly better than the performance of local radial basis function however computational cost of the local radial basis function is less than the computational cost of barycentric rational interpolation.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ömer OruçAbstractIn this paper, 2D viscoelastic wave equation is solved numerically both on regular and irregular domains. For spatial approximation of viscoelastic wave equation two meshless methods based on local radial basis function and barycentric rational interpolation are proposed. Both of the spatial approximation methods do not need mesh, node connectivity or integration process on local subdomains so they are truly meshless. For local radial basis function method we used an existing algorithm in literature to choose an acceptable shape parameter. Time marching is performed with fourth order Runge Kutta method. L2 and L∞ error norms for some test problems are reckoned to indicate efficiency and performance of the proposed two methods. Also, stability of the methods is discussed. Acquired results confirm the applicability of the proposed methods for 2D viscoelastic wave equation. We have performed some comparisons between the proposed two methods in the sense of accuracy and computational cost. From the comparisons, we have observed that performance of the barycentric rational interpolation in the sense of accuracy is slightly better than the performance of local radial basis function however computational cost of the local radial basis function is less than the computational cost of barycentric rational interpolation.
 Interaction between free surface flow and moving bodies with a dynamic
mesh and interface geometric reconstruction approach Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): MingJian LiAbstractTo investigate the fluidrigid body interaction issues with free surfaces, a numerical approach has been developed. This algorithm is in an arbitrary Lagrangian–Eulerian description and Volume of Fluid (VOF) framework, using dynamic unstructured mesh to solve the coupled system. The fluid–solid interface uses partitioned Dirichlet–Neumann iterations with Aitken’s relaxation. For the twophase fluids part, an interface geometric reconstruction approach has been applied to accurately capture the free surfaces. This piecewise linear interface calculation (PLIC) based method uses Newton’s iteration to efficiently reconstruct interfaces on an unstructured mesh, and applies an unsplit scheme to transport variables. The algorithm has been successfully implemented in open source code OpenFOAM®, and was compared with the latter’s builtin solver using interface compression method to deal with free surfaces. Numerical results suggest that our solver has better accuracy on multiphase flow problems, while the previous solver fails to obtain correct interfaces. Moreover, the capacity of accurately solving fluidrigid body interaction problems with free surfaces has been achieved. Validation cases are provided for fluid–structure interaction problems with and without free surfaces, and results are in accordance with analytical and experimental data from the literature. The algorithm and solver in this paper, can be applied on fluid–structure interaction cases with free surfaces in the future, such as sloshing and water entry problems.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): MingJian LiAbstractTo investigate the fluidrigid body interaction issues with free surfaces, a numerical approach has been developed. This algorithm is in an arbitrary Lagrangian–Eulerian description and Volume of Fluid (VOF) framework, using dynamic unstructured mesh to solve the coupled system. The fluid–solid interface uses partitioned Dirichlet–Neumann iterations with Aitken’s relaxation. For the twophase fluids part, an interface geometric reconstruction approach has been applied to accurately capture the free surfaces. This piecewise linear interface calculation (PLIC) based method uses Newton’s iteration to efficiently reconstruct interfaces on an unstructured mesh, and applies an unsplit scheme to transport variables. The algorithm has been successfully implemented in open source code OpenFOAM®, and was compared with the latter’s builtin solver using interface compression method to deal with free surfaces. Numerical results suggest that our solver has better accuracy on multiphase flow problems, while the previous solver fails to obtain correct interfaces. Moreover, the capacity of accurately solving fluidrigid body interaction problems with free surfaces has been achieved. Validation cases are provided for fluid–structure interaction problems with and without free surfaces, and results are in accordance with analytical and experimental data from the literature. The algorithm and solver in this paper, can be applied on fluid–structure interaction cases with free surfaces in the future, such as sloshing and water entry problems.
 A complex variable boundary point interpolation method for the nonlinear
Signorini problem Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Xiaolin Li, Shuling LiAbstractA complex variable boundary point interpolation method (CVBPIM) is presented for numerical analysis of the nonlinear Signorini problem. To reduce the computational cost and to improve the stability of the point interpolation method, a complex variable point interpolation method (CVPIM) is developed to construct meshless shape functions with interpolation properties. By using a projection technique to deal with the nonlinear inequality boundary conditions, and combining boundary integral equations with the CVPIM to establish discrete linear algebraic systems, the CVBPIM is an easytoimplement and boundaryonly meshless method for nonlinear Signorini problems. Numerical results are presented to show the accuracy and efficiency of the method.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Xiaolin Li, Shuling LiAbstractA complex variable boundary point interpolation method (CVBPIM) is presented for numerical analysis of the nonlinear Signorini problem. To reduce the computational cost and to improve the stability of the point interpolation method, a complex variable point interpolation method (CVPIM) is developed to construct meshless shape functions with interpolation properties. By using a projection technique to deal with the nonlinear inequality boundary conditions, and combining boundary integral equations with the CVPIM to establish discrete linear algebraic systems, the CVBPIM is an easytoimplement and boundaryonly meshless method for nonlinear Signorini problems. Numerical results are presented to show the accuracy and efficiency of the method.
 A shear stress transport incorporated elliptic blending turbulence model
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): X.L. Yang, Y. Liu, L. YangAbstractAn elliptic blending turbulence model, integrating the Shear Stress Transport (SST) characteristics in boundary layer together, is developed and validated. This model consists of four governing equations which have the same forms as those used in our previous kωφα model (belonging to the elliptic blending turbulence models). The major improvement is that, a new turbulent viscosity definition is constructed which inherits the advantages of the elliptic blending turbulence models and the SST turbulence models. The new model is applied to nearwall, separated and impinging jet flows and associated heat transfer problems. The results are compared with experimental and DNS data. Comparisons with the results of using the previously developed kωφα model and the Menter’s SST kω model are also carried out. It is shown that the current new model has similar behaviors with the previously developed kωφα model for the near wall flow and heat transfer problems. For separated and impinging jet flows and the associated heat transfer problems, the current new model yields better results than the SST kω model and our previous kωφα model.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): X.L. Yang, Y. Liu, L. YangAbstractAn elliptic blending turbulence model, integrating the Shear Stress Transport (SST) characteristics in boundary layer together, is developed and validated. This model consists of four governing equations which have the same forms as those used in our previous kωφα model (belonging to the elliptic blending turbulence models). The major improvement is that, a new turbulent viscosity definition is constructed which inherits the advantages of the elliptic blending turbulence models and the SST turbulence models. The new model is applied to nearwall, separated and impinging jet flows and associated heat transfer problems. The results are compared with experimental and DNS data. Comparisons with the results of using the previously developed kωφα model and the Menter’s SST kω model are also carried out. It is shown that the current new model has similar behaviors with the previously developed kωφα model for the near wall flow and heat transfer problems. For separated and impinging jet flows and the associated heat transfer problems, the current new model yields better results than the SST kω model and our previous kωφα model.

L ∞  Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Hailong Ye, Qiang Liu, Mingjun ZhouAbstractRecently, some fractional Cahn–Hilliard equations are proposed for phase transition and image process, etc., which have attracted a lot of attention. In this paper, we concern the L∞ bound of the solutions to a class of fractional Cahn–Hilliard equations, which extends the results of integer order. By an invariant derivative technique, the crucial uniform estimates related to the kernel function of fractional Cahn–Hilliard are established. The technique presented here can be also applied to the other related fractional models.
 Abstract: Publication date: Available online 17 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Hailong Ye, Qiang Liu, Mingjun ZhouAbstractRecently, some fractional Cahn–Hilliard equations are proposed for phase transition and image process, etc., which have attracted a lot of attention. In this paper, we concern the L∞ bound of the solutions to a class of fractional Cahn–Hilliard equations, which extends the results of integer order. By an invariant derivative technique, the crucial uniform estimates related to the kernel function of fractional Cahn–Hilliard are established. The technique presented here can be also applied to the other related fractional models.
 juSFEM: A Juliabased opensource package of parallel Smoothed Finite
Element Method (SFEM) for elastic problems Abstract: Publication date: Available online 15 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Zenan Huo, Gang Mei, Nengxiong XuAbstractThe Smoothed Finite Element Method (SFEM) proposed by Liu G.R. can achieve more accurate results than the conventional FEM. Currently, much commercial software and many opensource packages have been developed to analyze various science and engineering problems using the FEM. However, there is little work focusing on designing and developing software or packages for the SFEM. In this paper, we design and implement an opensource package of the parallel SFEM for elastic problems by utilizing the Julia language on multicore CPU. The Julia language is a fast, easytouse, and opensource programming language that was originally designed for highperformance computing. We term our package as juSFEM. To the best of the authors’ knowledge, juSFEM is the first package of parallel SFEM developed with the Julia language. To verify the correctness and evaluate the efficiency of juSFEM, two groups of benchmark tests are conducted. The benchmark results show that (1) juSFEM can achieve accurate results when compared to commercial FEM software ABAQUS, and (2) juSFEM only requires 543 s to calculate the displacements of a 3D elastic cantilever beam model which is composed of approximately 2 million tetrahedral elements, while in contrast the commercial FEM software needs 930 s for the same calculation model; (3) the parallel juSFEM executed on the 24core CPU is approximately 20× faster than the corresponding serial version. Moreover, the structure and function of juSFEM are easily modularized, and the code in juSFEM is clear and readable, which is convenient for further development.
 Abstract: Publication date: Available online 15 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Zenan Huo, Gang Mei, Nengxiong XuAbstractThe Smoothed Finite Element Method (SFEM) proposed by Liu G.R. can achieve more accurate results than the conventional FEM. Currently, much commercial software and many opensource packages have been developed to analyze various science and engineering problems using the FEM. However, there is little work focusing on designing and developing software or packages for the SFEM. In this paper, we design and implement an opensource package of the parallel SFEM for elastic problems by utilizing the Julia language on multicore CPU. The Julia language is a fast, easytouse, and opensource programming language that was originally designed for highperformance computing. We term our package as juSFEM. To the best of the authors’ knowledge, juSFEM is the first package of parallel SFEM developed with the Julia language. To verify the correctness and evaluate the efficiency of juSFEM, two groups of benchmark tests are conducted. The benchmark results show that (1) juSFEM can achieve accurate results when compared to commercial FEM software ABAQUS, and (2) juSFEM only requires 543 s to calculate the displacements of a 3D elastic cantilever beam model which is composed of approximately 2 million tetrahedral elements, while in contrast the commercial FEM software needs 930 s for the same calculation model; (3) the parallel juSFEM executed on the 24core CPU is approximately 20× faster than the corresponding serial version. Moreover, the structure and function of juSFEM are easily modularized, and the code in juSFEM is clear and readable, which is convenient for further development.
 A Petrov–Galerkin finite element method for simulating chemotaxis models
on stationary surfaces Abstract: Publication date: Available online 14 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Shubo Zhao, Xufeng Xiao, Jianping Zhao, Xinlong FengAbstractIn this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowingup problems and pattern formulations demonstrate the effectiveness of the proposed method.
 Abstract: Publication date: Available online 14 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Shubo Zhao, Xufeng Xiao, Jianping Zhao, Xinlong FengAbstractIn this paper, we present a Petrov–Galerkin finite element method for a class of chemotaxis models defined on surfaces, which describe the movement by one community in reaction to one chemical or biological signal on manifolds. It is desired for numerical methods to satisfy discrete maximum principle and discrete mass conservation property, which is a challenge due to the singular behavior of numerical solution. Thus a Petrov–Galerkin method is combined with an effective mass conservation factor to overcome the challenge. Furthermore, we prove two facts, this method maintains positivity and discrete mass conservation property. In addition, decoupled approach is applied based on the gradient and Laplacian recoveries to solve the coupling system. The relevant stability analyses is provided. Finally, numerical simulations of blowingup problems and pattern formulations demonstrate the effectiveness of the proposed method.
 A multigrid–homotopy method for nonlinear inverse problems
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Tao LiuAbstractIn the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of subinverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the twophase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Tao LiuAbstractIn the present contribution, we develop a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of subinverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection–diffusion equation, which is the saturation equation within the twophase porous media flow. We demonstrate the effectiveness and merits of the multigrid–homotopy method on two actual model problems.
 A RBFbased technique for 3D convection–diffusion–reaction problems in
an anisotropic inhomogeneous medium Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Sergiy Reutskiy, Ji LinAbstractWe present a RBFbased semianalytical technique for solving 3D convection–diffusion–reaction (CDR) equations to model transport in an anisotropic inhomogeneous medium. The mathematical model is expressed as the boundary value problem for elliptic partial differential equation (EPDE). Main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the EPDE inside. To be more precise, we transform the original EPDE to the equation with homogeneous boundary condition (BC) and seek the approximate solution as a sum of the modified RBFs (MRBFs). The MRBFs satisfy the homogeneous BC of the problem. So, any linear combination also satisfies the homogeneous BC. The RBFs of three types are used in the framework of the method: the Multiquadric (MQ) RBF, the Gaussian RBF and the conical one. The coefficients of the linear combination are determined so that it satisfies the governing equation of the EPDE. Ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3D CDR problems in single and double connected domains.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Sergiy Reutskiy, Ji LinAbstractWe present a RBFbased semianalytical technique for solving 3D convection–diffusion–reaction (CDR) equations to model transport in an anisotropic inhomogeneous medium. The mathematical model is expressed as the boundary value problem for elliptic partial differential equation (EPDE). Main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the EPDE inside. To be more precise, we transform the original EPDE to the equation with homogeneous boundary condition (BC) and seek the approximate solution as a sum of the modified RBFs (MRBFs). The MRBFs satisfy the homogeneous BC of the problem. So, any linear combination also satisfies the homogeneous BC. The RBFs of three types are used in the framework of the method: the Multiquadric (MQ) RBF, the Gaussian RBF and the conical one. The coefficients of the linear combination are determined so that it satisfies the governing equation of the EPDE. Ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3D CDR problems in single and double connected domains.
 Mathematical modelling of biodegradation in situ application to
biodenitrification Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Mostafa Abaali, Zoubida MghazliAbstractIn the biological units in situ, there is proved the existence of free and adherent bacteria. Most mathematical models deal with only free or adherent bacteria, and do not consider the bacteria diffusion and transport. In this work we consider a mathematical model of biodenitrifcation taking into account free and adherent bacteria, their interdependence and the diffusion and transport related to them. The model is presented first in a reactor and then in a porous medium. The equations obtained are approximated by a Finite Element method. The numerical tests presented are close to experimental results in the literature, that confirm the validity of the model, and show the importance of considering the complete model taking into account both types of bacteria and their evolution in time and space.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Mostafa Abaali, Zoubida MghazliAbstractIn the biological units in situ, there is proved the existence of free and adherent bacteria. Most mathematical models deal with only free or adherent bacteria, and do not consider the bacteria diffusion and transport. In this work we consider a mathematical model of biodenitrifcation taking into account free and adherent bacteria, their interdependence and the diffusion and transport related to them. The model is presented first in a reactor and then in a porous medium. The equations obtained are approximated by a Finite Element method. The numerical tests presented are close to experimental results in the literature, that confirm the validity of the model, and show the importance of considering the complete model taking into account both types of bacteria and their evolution in time and space.
 A virtual element method for stochastic Stokes equations
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Wei Liu, Huoyuan DuanAbstractIn this paper, a virtual element method for the stochastic Stokes equations driven by an additive white noise is proposed and analyzed. The velocity is approximated by the lowestorder virtual element which is originally designed for the Poisson equation and the projection is also taken as the one originally for the Poisson equation, while the pressure is approximated by the traditional discontinuous piecewise constant element. For stable approximations, we adopt a stabilization associating with the pressure jumps. We show the infsup condition and derive the stability. We moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the Green function. Numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Wei Liu, Huoyuan DuanAbstractIn this paper, a virtual element method for the stochastic Stokes equations driven by an additive white noise is proposed and analyzed. The velocity is approximated by the lowestorder virtual element which is originally designed for the Poisson equation and the projection is also taken as the one originally for the Poisson equation, while the pressure is approximated by the traditional discontinuous piecewise constant element. For stable approximations, we adopt a stabilization associating with the pressure jumps. We show the infsup condition and derive the stability. We moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the Green function. Numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained.

q Gaussian+process&rft.title=Computers+&+Mathematics+with+Applications&rft.issn=08981221&rft.date=&rft.volume=">The nonMarkovian property of q Gaussian process Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): LiMin Liu, YingYing Cui, Jie Xu, Chao Li, QingHui GaoAbstractIn this paper, the qGaussian process based on the nonextensive theory is discussed from a mathematical point of view, which has been widely applied to many anomalous diffusion systems in physics and finance. Firstly, the discussion of nonMarkovian property of qGaussian process provides a numerical support for the future theoretical research. Secondly, the martingale and selfsimilarity of this process are obtained by Tsallis distributions. Thirdly, the long dependence is analyzed by simulations and Hurst exponents are compared with those of fractional Brownian motion. At last, the European call option price formula driven by this process is simulated, by which we find that this process can better match anomalous diffusion and the volatility smile.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): LiMin Liu, YingYing Cui, Jie Xu, Chao Li, QingHui GaoAbstractIn this paper, the qGaussian process based on the nonextensive theory is discussed from a mathematical point of view, which has been widely applied to many anomalous diffusion systems in physics and finance. Firstly, the discussion of nonMarkovian property of qGaussian process provides a numerical support for the future theoretical research. Secondly, the martingale and selfsimilarity of this process are obtained by Tsallis distributions. Thirdly, the long dependence is analyzed by simulations and Hurst exponents are compared with those of fractional Brownian motion. At last, the European call option price formula driven by this process is simulated, by which we find that this process can better match anomalous diffusion and the volatility smile.
 Coupled simulation of transient heat flow and electric currents in thin
wires: Application to bond wires in microelectronic chip packaging Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Thorben Casper, Ulrich Römer, Herbert De Gersem, Sebastian SchöpsAbstractThis work addresses the simulation of heat flow and electric currents in thin wires. An important application is the use of bond wires in microelectronic chip packaging. The heat distribution is modeled by an electrothermal coupled problem, which poses numerical challenges due to the presence of different geometric scales. The necessity of very fine grids is relaxed by solving and embedding a 1D subproblem along the wire into the surrounding 3D geometry. The arising singularities are described using de Rham currents. It is shown that the problem is related to fluid flow in porous 3D media with 1D fractures (C. D’Angelo, SIAM Journal of Numerical Analysis 50.1, pp. 194–215, 2012). A careful formulation of the 1D–3D coupling condition is essential to obtain a stable scheme that yields a physical solution. Elliptic model problems are used to investigate the numerical errors and the corresponding convergence rates. Additionally, the transient electrothermal simulation of a simplified microelectronic chip package as used in industrial applications is presented.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Thorben Casper, Ulrich Römer, Herbert De Gersem, Sebastian SchöpsAbstractThis work addresses the simulation of heat flow and electric currents in thin wires. An important application is the use of bond wires in microelectronic chip packaging. The heat distribution is modeled by an electrothermal coupled problem, which poses numerical challenges due to the presence of different geometric scales. The necessity of very fine grids is relaxed by solving and embedding a 1D subproblem along the wire into the surrounding 3D geometry. The arising singularities are described using de Rham currents. It is shown that the problem is related to fluid flow in porous 3D media with 1D fractures (C. D’Angelo, SIAM Journal of Numerical Analysis 50.1, pp. 194–215, 2012). A careful formulation of the 1D–3D coupling condition is essential to obtain a stable scheme that yields a physical solution. Elliptic model problems are used to investigate the numerical errors and the corresponding convergence rates. Additionally, the transient electrothermal simulation of a simplified microelectronic chip package as used in industrial applications is presented.
 A generalized lattice Boltzmann model for fluid flow system and its
application in twophase flows Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Xiaolei Yuan, Zhenhua Chai, Huili Wang, Baochang ShiAbstractIn this paper, a generalized lattice Boltzmann (LB) model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible Navier–Stokes (N–S) equations. This model can be used to deal with singlephase and twophase flows problems with a source term in the continuity equation. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a twophase flow system, and in the frame work of our generalized LB model, a new phasefieldbased LB model is developed for incompressible and quasiincompressible twophase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasiincompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of twophase flows. Finally, we also conduct a comparison between quasiincompressible and incompressible LB models for twophase flow problems, and find that in some cases, the proposed quasiincompressible LB model performs better than incompressible LB models.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Xiaolei Yuan, Zhenhua Chai, Huili Wang, Baochang ShiAbstractIn this paper, a generalized lattice Boltzmann (LB) model with a source term in the continuity equation is proposed to solve both incompressible and nearly incompressible Navier–Stokes (N–S) equations. This model can be used to deal with singlephase and twophase flows problems with a source term in the continuity equation. From this generalized model, we can not only get some existing models, but also derive new models. Moreover, for the incompressible model derived, a modified pressure scheme is introduced to calculate the pressure, and then to ensure the accuracy of the model. In this work, we will focus on a twophase flow system, and in the frame work of our generalized LB model, a new phasefieldbased LB model is developed for incompressible and quasiincompressible twophase flows. A series of numerical simulations of some classic physical problems, including a spinodal decomposition, a static droplet, a layered Poiseuille flow, and a bubble rising flow under buoyancy, are performed to validate the developed model. Besides, some comparisons with previous quasiincompressible and incompressible LB models are also carried out, and the results show that the present model is accurate in the study of twophase flows. Finally, we also conduct a comparison between quasiincompressible and incompressible LB models for twophase flow problems, and find that in some cases, the proposed quasiincompressible LB model performs better than incompressible LB models.
 A splitting Fourier pseudospectral method for
Vlasov–Poisson–Fokker–Planck system Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Hanquan Wang, Ronghua Cheng, Xinming WuAbstractIn this paper, we propose a splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck (VPFP) system, which describes the motion of charged particles in plasma. The numerical integration for the system is performed by employing the splitting method in time, Fourier Galerkin method in space direction, and Fourier collocation method in phase direction, respectively. The algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast Fourier transform and technique of diagonalization, respectively. Extensive numerical results in onedimensional phase space (or 1x×1v) from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies. Numerical algorithm for the VPFP system in twodimensional phase space (or 2x×2v) has been summarized in Appendix.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Hanquan Wang, Ronghua Cheng, Xinming WuAbstractIn this paper, we propose a splitting Fourier pseudospectral method for Vlasov–Poisson–Fokker–Planck (VPFP) system, which describes the motion of charged particles in plasma. The numerical integration for the system is performed by employing the splitting method in time, Fourier Galerkin method in space direction, and Fourier collocation method in phase direction, respectively. The algorithm has spectral accuracy in both space and phase directions and can be implemented efficiently with the fast Fourier transform and technique of diagonalization, respectively. Extensive numerical results in onedimensional phase space (or 1x×1v) from the proposed numerical method are shown and have proven the good agreement with the theory and previous studies. Numerical algorithm for the VPFP system in twodimensional phase space (or 2x×2v) has been summarized in Appendix.
 New applications of numerical simulation based on lattice Boltzmann method
at high Reynolds numbers Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Bo An, J.M. Bergadà, F. Mellibovsky, W.M. SangAbstractIn order to study the flow behavior at high Reynolds numbers, two modified models, known as the multiplerelaxationtime lattice Boltzmann method (MRTLBM) and largeeddysimulation lattice Boltzmann method (LESLBM), have been employed in this paper. The MRTLBM was designed to improve numerical stability at high Reynolds numbers, by introducing multiple relaxation time terms, which consider the variations of density, energy, momentum, energy flux and viscous stress tensor. As a result, MRTLBM is capable of dealing with turbulent flows considering energy dispersion and dissipation. In the present paper, this model was employed to simulate the flow at turbulent Reynolds numbers in walldriven cavities. Twosided wall driven cavity flow was studied for the first time, based on MRTLBM, at Reynolds numbers ranging from 2×104to1×106, and employing a very large resolution2048 × 2048. It is found that whenever top and bottom lids are moving in the opposite directions, and the Reynolds number is higher than 2×104, the flow is chaotic, although some quasisymmetric properties still remain, fully disappearing at Reynolds numbers between 2×105 and 3×105. Furthermore, between this Reynolds numbers range, 2×105
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Bo An, J.M. Bergadà, F. Mellibovsky, W.M. SangAbstractIn order to study the flow behavior at high Reynolds numbers, two modified models, known as the multiplerelaxationtime lattice Boltzmann method (MRTLBM) and largeeddysimulation lattice Boltzmann method (LESLBM), have been employed in this paper. The MRTLBM was designed to improve numerical stability at high Reynolds numbers, by introducing multiple relaxation time terms, which consider the variations of density, energy, momentum, energy flux and viscous stress tensor. As a result, MRTLBM is capable of dealing with turbulent flows considering energy dispersion and dissipation. In the present paper, this model was employed to simulate the flow at turbulent Reynolds numbers in walldriven cavities. Twosided wall driven cavity flow was studied for the first time, based on MRTLBM, at Reynolds numbers ranging from 2×104to1×106, and employing a very large resolution2048 × 2048. It is found that whenever top and bottom lids are moving in the opposite directions, and the Reynolds number is higher than 2×104, the flow is chaotic, although some quasisymmetric properties still remain, fully disappearing at Reynolds numbers between 2×105 and 3×105. Furthermore, between this Reynolds numbers range, 2×105
 A generalization of trigonometric transform splitting methods for spatial
fractional diffusion equations Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): XinHui Shao, ZhenDuo Zhang, HaiLong ShenAbstractThe spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted Grünwald formula and the given discretized linear systems have a diagonalplusToeplitz structure. In this paper, based on trigonometric transformation splitting (TTS), we study efficient iterative method called GTTS has two parameters. As a focus, we give the simple and effective optimal forms of these two parameters respectively. We carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): XinHui Shao, ZhenDuo Zhang, HaiLong ShenAbstractThe spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted Grünwald formula and the given discretized linear systems have a diagonalplusToeplitz structure. In this paper, based on trigonometric transformation splitting (TTS), we study efficient iterative method called GTTS has two parameters. As a focus, we give the simple and effective optimal forms of these two parameters respectively. We carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm.
 On the application of isogeometric finite volume method in numerical
analysis of wetsteam flow through turbine cascades Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Ali Hashemian, Esmail Lakzian, Amir EbrahimiFizikAbstractThe isogeometric finite volume analysis is utilized in this research to numerically simulate the twodimensional viscous wetsteam flow between stationary cascades of a steam turbine for the first time. In this approach, the analysissuitable computational mesh with “curved” boundaries is generated for the fluid flow by employing a nonuniform rational Bspline (NURBS) surface that describes the cascade geometry, and the governing equations are then discretized by the NURBS representation. Thanks to smooth and accurate geometry representation of the NURBS formulation, the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems, but also, as validated against wellestablished experiments, significantly improves the accuracy of the numerical solution. In addition, the shock location in the cascade is predicted and tracked with a sufficient accuracy.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Ali Hashemian, Esmail Lakzian, Amir EbrahimiFizikAbstractThe isogeometric finite volume analysis is utilized in this research to numerically simulate the twodimensional viscous wetsteam flow between stationary cascades of a steam turbine for the first time. In this approach, the analysissuitable computational mesh with “curved” boundaries is generated for the fluid flow by employing a nonuniform rational Bspline (NURBS) surface that describes the cascade geometry, and the governing equations are then discretized by the NURBS representation. Thanks to smooth and accurate geometry representation of the NURBS formulation, the employed isogeometric framework not only resolves issues concerning the conventional mesh generation techniques of the finite volume method in steam turbine problems, but also, as validated against wellestablished experiments, significantly improves the accuracy of the numerical solution. In addition, the shock location in the cascade is predicted and tracked with a sufficient accuracy.
 DirichlettoNeumann boundary conditions for multiple scattering in
waveguides Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Youngho Min, Seungil KimAbstractIn this paper, we study a multiple DirichlettoNeumann (MDtN) boundary condition for solving a timeharmonic multiple scattering problem governed by the Helmholtz equation in waveguides that include multiple obstacles, cavities or inhomogeneities with straight waveguides placed between them. The MDtN condition is derived by analyzing analytic solutions represented by Fourier series in the straight waveguides between obstacles, cavities or inhomogeneities. The proposed method is then to remove the straight waveguides between scatterers and impose the MDtN condition on artificial boundaries resulting from domain truncation. This numerical technique can allow a great reduction of computational efforts. The wellposedness of the reduced problem with the full MDtN condition and the reduced problem with truncated MDtN conditions are established. Also the exponential convergence of approximate solutions satisfying truncated MDtN conditions will be proved.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Youngho Min, Seungil KimAbstractIn this paper, we study a multiple DirichlettoNeumann (MDtN) boundary condition for solving a timeharmonic multiple scattering problem governed by the Helmholtz equation in waveguides that include multiple obstacles, cavities or inhomogeneities with straight waveguides placed between them. The MDtN condition is derived by analyzing analytic solutions represented by Fourier series in the straight waveguides between obstacles, cavities or inhomogeneities. The proposed method is then to remove the straight waveguides between scatterers and impose the MDtN condition on artificial boundaries resulting from domain truncation. This numerical technique can allow a great reduction of computational efforts. The wellposedness of the reduced problem with the full MDtN condition and the reduced problem with truncated MDtN conditions are established. Also the exponential convergence of approximate solutions satisfying truncated MDtN conditions will be proved.
 Automatic prior shape selection for image edge detection with modified
Mumford–Shah model Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Yuying Shi, Zhimei Huo, Jing Qin, Yilin LiAbstractEdge detection plays an important role in the field of image processing. In this paper, we propose a novel variational model to automatically and adaptively detect one or more prior shapes from the given dictionary to guide the edge detection process. In that way, we can effectively detect the shapes of interest from the test image. Moreover, an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) is proposed to solve this model with guaranteed convergence. A variety of numerical experiments show that the proposed method has achieved ideal performance for edge detection in images with missing information, various types of noise and complicated background, and even multiple objects.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Yuying Shi, Zhimei Huo, Jing Qin, Yilin LiAbstractEdge detection plays an important role in the field of image processing. In this paper, we propose a novel variational model to automatically and adaptively detect one or more prior shapes from the given dictionary to guide the edge detection process. In that way, we can effectively detect the shapes of interest from the test image. Moreover, an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) is proposed to solve this model with guaranteed convergence. A variety of numerical experiments show that the proposed method has achieved ideal performance for edge detection in images with missing information, various types of noise and complicated background, and even multiple objects.
 Polynomially bounded error estimates for Trapezoidal Rule Convolution
Quadrature Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Hasan Eruslu, FranciscoJavier SayasAbstractWe clarify the dependence with respect to the time variable of some estimates about the convergence of the Trapezoidal Rule based Convolution Quadrature method applied to hyperbolic problems. This requires a careful investigation of the article of Lehel Banjai where the first convergence estimates were introduced, and of some technical results from a classical paper of Christian Lubich.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Hasan Eruslu, FranciscoJavier SayasAbstractWe clarify the dependence with respect to the time variable of some estimates about the convergence of the Trapezoidal Rule based Convolution Quadrature method applied to hyperbolic problems. This requires a careful investigation of the article of Lehel Banjai where the first convergence estimates were introduced, and of some technical results from a classical paper of Christian Lubich.
 A numerical study of the higherdimensional GelfandBratu model
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Sehar Iqbal, Paul Andries ZegelingAbstractIn this article, a higher dimensional nonlinear boundaryvalue problem, viz., GelfandBratu (GB) problem, is solved numerically. For the threedimensional case, we present an accurate and efficient nonlinear multigrid (MG) approach and investigate multiplicities depending on the bifurcation parameter λ. We adopt a nonlinear MG approach Full approximation scheme (FAS) extended with a Krylov method as a smoother to handle the computational difficulties for obtaining the upper branches of the solutions. Further, we examine the numerical bifurcation behaviour of the GB problem in 3D and identify the existence of two new bifurcation points. Experiments illustrate the convergence of the numerical solutions and demonstrate the effectiveness of the proposed numerical strategy for all parameter values λ∈(0,λc]. For higher dimensions, we transform the GB problem, using ndimensional spherical coordinates, to a nonlinear ordinary differential equation (ODE). The numerical solutions of this nonlinear ODE are computed by a shooting method for a range of values of the dimension parameter n. Numerical experiments show the existence of several types of solutions for different values of n and λ. These results confirm the bifurcation behaviour of the higher dimensional GB problem as predicted from theoretical results in literature.
 Abstract: Publication date: 15 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 6Author(s): Sehar Iqbal, Paul Andries ZegelingAbstractIn this article, a higher dimensional nonlinear boundaryvalue problem, viz., GelfandBratu (GB) problem, is solved numerically. For the threedimensional case, we present an accurate and efficient nonlinear multigrid (MG) approach and investigate multiplicities depending on the bifurcation parameter λ. We adopt a nonlinear MG approach Full approximation scheme (FAS) extended with a Krylov method as a smoother to handle the computational difficulties for obtaining the upper branches of the solutions. Further, we examine the numerical bifurcation behaviour of the GB problem in 3D and identify the existence of two new bifurcation points. Experiments illustrate the convergence of the numerical solutions and demonstrate the effectiveness of the proposed numerical strategy for all parameter values λ∈(0,λc]. For higher dimensions, we transform the GB problem, using ndimensional spherical coordinates, to a nonlinear ordinary differential equation (ODE). The numerical solutions of this nonlinear ODE are computed by a shooting method for a range of values of the dimension parameter n. Numerical experiments show the existence of several types of solutions for different values of n and λ. These results confirm the bifurcation behaviour of the higher dimensional GB problem as predicted from theoretical results in literature.
 Convergence of finite element methods for hyperbolic heat conduction model
with an interface Abstract: Publication date: Available online 10 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Bhupen Deka, Jogen DuttaAbstractThe paper concerns numerical study of nonFourier bio heat transfer model in multilayered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for nonhomogeneous inner structures.
 Abstract: Publication date: Available online 10 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Bhupen Deka, Jogen DuttaAbstractThe paper concerns numerical study of nonFourier bio heat transfer model in multilayered media. Specifically, we employ the Maxwell–Cattaneo equation on the physical media with discontinuous coefficients. A fitted finite element method is proposed and analyzed for a hyperbolic heat conduction model with discontinuous coefficients. Typical semidiscrete and fully discrete schemes are presented for a fitted finite element discretization with straight interface triangles. The fully discrete space–time finite element discretizations are based on second order in time Newmark scheme. Optimal a priori error estimates for both semidiscrete and fully discrete schemes are proved in L∞(L2) norm. Numerical experiments are reported for several test cases to confirm our theoretical convergence rate. Finite element algorithm presented here can be used to solve a wide variety of hyperbolic heat conduction models for nonhomogeneous inner structures.
 Convergence of the optimality criteria method for multiple state optimal
design problems Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Krešimir Burazin, Ivana CrnjacAbstractWe consider multiple state optimal design problems with two isotropic materials from the conductivity point of view. Since the classical solutions of these problems usually do not exist, a proper relaxation of the original problem is obtained, using the homogenization method. In [1] we derive necessary conditions of optimality of the relaxed problem, which enables us to implement a new variant of the optimality criteria method. It appears that this variant gives converging sequence of designs for the energy minimization problems. In this work we prove convergence of the method for energy minimization problems in the spherically symmetric case and in a case when the number of states is less than the space dimension.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Krešimir Burazin, Ivana CrnjacAbstractWe consider multiple state optimal design problems with two isotropic materials from the conductivity point of view. Since the classical solutions of these problems usually do not exist, a proper relaxation of the original problem is obtained, using the homogenization method. In [1] we derive necessary conditions of optimality of the relaxed problem, which enables us to implement a new variant of the optimality criteria method. It appears that this variant gives converging sequence of designs for the energy minimization problems. In this work we prove convergence of the method for energy minimization problems in the spherically symmetric case and in a case when the number of states is less than the space dimension.
 A simplified finite volume lattice Boltzmann method for simulations of
fluid flows from laminar to turbulent regime, Part I: Numerical framework
and its application to laminar flow simulation Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yong Wang, Chengwen Zhong, Jun Cao, Congshan Zhuo, Sha LiuAbstractIn this paper, an unstructured grid based finite volume lattice Boltzmann method (FVLBM) that can be used for the simulation of incompressible laminar flows is presented and studied in detail. This method is derived from a simple modification of the cellvertex unstructured grid based FVLBM proposed by Stiebler et al. (2006). Compared with other complex flux reconstruction methods, the present scheme has a low computational cost and can also achieve secondorder spatial accuracy. Furthermore, depending on the use of the different temporal discretization schemes, the temporal accuracy can be adjusted for both steady and unsteady flows. Besides, some comparisons of the computational cost and accuracy with another FVLBM scheme are also presented. Meanwhile, different boundary conditions are illustrated that are easy to implement on complex geometries. To validate the present method, four cases are carried out, including a Couette flow driven by one plate for an accuracy test, flow in a square cavity, flow around a single circular cylinder and a more complex flow around double circular cylinders. Numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high Reynolds number with relatively few grid cells, thus demonstrating the good capability of the present method.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yong Wang, Chengwen Zhong, Jun Cao, Congshan Zhuo, Sha LiuAbstractIn this paper, an unstructured grid based finite volume lattice Boltzmann method (FVLBM) that can be used for the simulation of incompressible laminar flows is presented and studied in detail. This method is derived from a simple modification of the cellvertex unstructured grid based FVLBM proposed by Stiebler et al. (2006). Compared with other complex flux reconstruction methods, the present scheme has a low computational cost and can also achieve secondorder spatial accuracy. Furthermore, depending on the use of the different temporal discretization schemes, the temporal accuracy can be adjusted for both steady and unsteady flows. Besides, some comparisons of the computational cost and accuracy with another FVLBM scheme are also presented. Meanwhile, different boundary conditions are illustrated that are easy to implement on complex geometries. To validate the present method, four cases are carried out, including a Couette flow driven by one plate for an accuracy test, flow in a square cavity, flow around a single circular cylinder and a more complex flow around double circular cylinders. Numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high Reynolds number with relatively few grid cells, thus demonstrating the good capability of the present method.
 PDE models for American options with counterparty risk and two stochastic
factors: Mathematical analysis and numerical solution Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Iñigo Arregui, Beatriz Salvador, Daniel Ševčovič, Carlos VázquezAbstractIn this article we propose new linear and nonlinear partial differential equations (PDEs) models for pricing American options and total value adjustment in the presence of counterparty risk. An innovative aspect comes from the consideration of stochastic spreads, which increases the dimension of the problem. In this setting, we pose new complementarity problems associated to linear and nonlinear PDEs. Moreover, using the mathematical tools of semilinear variational inequalities for parabolic equations, we prove the existence and uniqueness of a solution for these models. For the numerical solution, we mainly combine a semiLagrangian time discretization scheme, a fixed point method to cope with nonlinear terms and a finite element method for the spatial discretization, jointly with an augmented Lagrangian active set method to solve the fully discretized complementarity problem. Finally, numerical examples illustrate the expected behaviour of the option prices and the corresponding total value adjustment, as well as the performance of the proposed numerical techniques. Moreover, we compare the numerical results from the PDEs approach with those obtained by applying Monte Carlo techniques.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Iñigo Arregui, Beatriz Salvador, Daniel Ševčovič, Carlos VázquezAbstractIn this article we propose new linear and nonlinear partial differential equations (PDEs) models for pricing American options and total value adjustment in the presence of counterparty risk. An innovative aspect comes from the consideration of stochastic spreads, which increases the dimension of the problem. In this setting, we pose new complementarity problems associated to linear and nonlinear PDEs. Moreover, using the mathematical tools of semilinear variational inequalities for parabolic equations, we prove the existence and uniqueness of a solution for these models. For the numerical solution, we mainly combine a semiLagrangian time discretization scheme, a fixed point method to cope with nonlinear terms and a finite element method for the spatial discretization, jointly with an augmented Lagrangian active set method to solve the fully discretized complementarity problem. Finally, numerical examples illustrate the expected behaviour of the option prices and the corresponding total value adjustment, as well as the performance of the proposed numerical techniques. Moreover, we compare the numerical results from the PDEs approach with those obtained by applying Monte Carlo techniques.
 Analysis of the parametric models of passive scalar transport used in the
lattice Boltzmann method Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Gerasim V. KrivovichevAbstractThe paper is devoted to the analysis of passive scalar transport models, used in the lattice Boltzmann method. The case of the pure advection process, without physical diffusion, is considered. The models, proposed by other authors, the modifications of these models and new highorder finitedifference schemes, based on the extended Runge–Kuttalike formulae, are analyzed. The attention is focused on the stability analysis, analysis of the fictitious numerical effects and on the investigation of the sensitivity of accuracy order to the parameter values.The stability analysis is based on the von Neumann method. The influence of the parameter values on the stability is demonstrated. Stability conditions are obtained. Numerical dispersion and diffusion are analyzed. Test problems with discontinuous and smooth initial conditions are considered. The sensitivity of the accuracy order on the parameter values is analyzed.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Gerasim V. KrivovichevAbstractThe paper is devoted to the analysis of passive scalar transport models, used in the lattice Boltzmann method. The case of the pure advection process, without physical diffusion, is considered. The models, proposed by other authors, the modifications of these models and new highorder finitedifference schemes, based on the extended Runge–Kuttalike formulae, are analyzed. The attention is focused on the stability analysis, analysis of the fictitious numerical effects and on the investigation of the sensitivity of accuracy order to the parameter values.The stability analysis is based on the von Neumann method. The influence of the parameter values on the stability is demonstrated. Stability conditions are obtained. Numerical dispersion and diffusion are analyzed. Test problems with discontinuous and smooth initial conditions are considered. The sensitivity of the accuracy order on the parameter values is analyzed.
 Unsteady interface boundary conditions for nearwall turbulence modeling
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): S. Utyuzhnikov, N. SmirnovaAbstractNumerical modeling nearwall turbulent flows is one of major challenges in the fluid dynamics. It inevitably requires significant computational resources. To overcome these difficulties, there are two principal ways based on simplification of the mathematical model and development of adhoc computational approaches. The nearwall nonoverlapping domain decomposition approach with the use of Robin interface boundary conditions proved to be very efficient. As has been shown, to apply this approach to essentially unsteady flows, the interface boundary conditions must be nonlocal in time and should be modified to include a memory term. In the current paper, it is proven that the memory term must be caused by the unsteadiness of both the solution at the interface boundary and driving force. The properties of the derived unsteady interface boundary conditions are studied in detail in the application to oscillatory and pulsating laminar flows in a channel and pipe. In particular, we study the effect of the memory term on the reproduction of the unsteady effects. The convergence to the exact solution is theoretically proven and numerically demonstrated. A practical calculation of the memory term is based on a Fourier expansion. It is also proven that the convergence is quadratic with respect to the inverse number of Fourier terms. A key question whether the nearwall domain decomposition can damage an external instability plays an important role to identify the perspectives of the technique to be extended to RANS/ LES (or DNS) decompositions.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): S. Utyuzhnikov, N. SmirnovaAbstractNumerical modeling nearwall turbulent flows is one of major challenges in the fluid dynamics. It inevitably requires significant computational resources. To overcome these difficulties, there are two principal ways based on simplification of the mathematical model and development of adhoc computational approaches. The nearwall nonoverlapping domain decomposition approach with the use of Robin interface boundary conditions proved to be very efficient. As has been shown, to apply this approach to essentially unsteady flows, the interface boundary conditions must be nonlocal in time and should be modified to include a memory term. In the current paper, it is proven that the memory term must be caused by the unsteadiness of both the solution at the interface boundary and driving force. The properties of the derived unsteady interface boundary conditions are studied in detail in the application to oscillatory and pulsating laminar flows in a channel and pipe. In particular, we study the effect of the memory term on the reproduction of the unsteady effects. The convergence to the exact solution is theoretically proven and numerically demonstrated. A practical calculation of the memory term is based on a Fourier expansion. It is also proven that the convergence is quadratic with respect to the inverse number of Fourier terms. A key question whether the nearwall domain decomposition can damage an external instability plays an important role to identify the perspectives of the technique to be extended to RANS/ LES (or DNS) decompositions.
 Singular functions for heterogeneous composites with cracks and notches;
the use of equilibrated singular basis functions Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): O. Bateniparvar, N. Noormohammadi, B. BoroomandAbstractIn this paper a novel method is presented to solve problems with weak singularities in twodimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other meshbased or meshless approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar coordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the wellknown literature or solved by wellestablished techniques, will demonstrate the capability of the method in problems related to composite materials.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): O. Bateniparvar, N. Noormohammadi, B. BoroomandAbstractIn this paper a novel method is presented to solve problems with weak singularities in twodimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other meshbased or meshless approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar coordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the wellknown literature or solved by wellestablished techniques, will demonstrate the capability of the method in problems related to composite materials.
 A structured approach to the construction of stable linear Lattice
Boltzmann collision operator Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Philipp Otte, Martin FrankAbstractWe introduce a structured approach to the construction of linear BGKtype collision operators ensuring that the resulting LatticeBoltzmann methods are stable with respect to a weighted L2norm. The results hold for particular boundary conditions including periodic, bounceback, and bounceback with flipping of sign boundary conditions. This construction uses the equivalent momentspace definition of BGKtype collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency. The presented structured approach is then applied to the 3D isothermal linearized Euler equations with nonvanishing background velocity. Finally, convergence results in the strong discrete L∞norm highlight the suitability of the structured approach introduced in this manuscript.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Philipp Otte, Martin FrankAbstractWe introduce a structured approach to the construction of linear BGKtype collision operators ensuring that the resulting LatticeBoltzmann methods are stable with respect to a weighted L2norm. The results hold for particular boundary conditions including periodic, bounceback, and bounceback with flipping of sign boundary conditions. This construction uses the equivalent momentspace definition of BGKtype collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency. The presented structured approach is then applied to the 3D isothermal linearized Euler equations with nonvanishing background velocity. Finally, convergence results in the strong discrete L∞norm highlight the suitability of the structured approach introduced in this manuscript.
 Symmetrical martingale solutions of backward doubly stochastic Volterra
integral equations Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Jiaqiang Wen, Yufeng ShiAbstractThis paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of symmetrical martingale solutions (SMsolutions, for short) is introduced for BDSVIEs. And the existence and uniqueness theorem for BDSVIEs in the sense of SMsolutions is established.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Jiaqiang Wen, Yufeng ShiAbstractThis paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of symmetrical martingale solutions (SMsolutions, for short) is introduced for BDSVIEs. And the existence and uniqueness theorem for BDSVIEs in the sense of SMsolutions is established.
 A parameterized deteriorated PSS preconditioner and its optimization for
nonsymmetric saddle point problems Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): JunLi Zhu, AiLi Yang, YuJiang WuAbstractBased on the deteriorated positive definite and skewHermitian splitting (DPSS) preconditioner, we develop a parameterized DPSS (PDPSS) preconditioner for the nonsymmetric saddle point linear systems. The spectral properties of the PDPSS preconditioned coefficient matrix are analyzed. Moreover, an upper bound of the degree of the minimal polynomial of the PDPSS preconditioned matrix is also obtained. Inasmuch as the efficiency of the PDPSS preconditioner depends on the values of its parameters, we further derive a fast and effective method to compute the optimal parameter values of the PDPSS preconditioner. Finally, numerical examples are employed to illustrate the feasibility and the efficiency of the PDPSS preconditioner.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): JunLi Zhu, AiLi Yang, YuJiang WuAbstractBased on the deteriorated positive definite and skewHermitian splitting (DPSS) preconditioner, we develop a parameterized DPSS (PDPSS) preconditioner for the nonsymmetric saddle point linear systems. The spectral properties of the PDPSS preconditioned coefficient matrix are analyzed. Moreover, an upper bound of the degree of the minimal polynomial of the PDPSS preconditioned matrix is also obtained. Inasmuch as the efficiency of the PDPSS preconditioner depends on the values of its parameters, we further derive a fast and effective method to compute the optimal parameter values of the PDPSS preconditioner. Finally, numerical examples are employed to illustrate the feasibility and the efficiency of the PDPSS preconditioner.
 Numerically pricing convertible bonds under stochastic volatility or
stochastic interest rate with an ADIbased predictor–corrector scheme Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Sha Lin, SongPing ZhuAbstractIn this paper, the pricing problem for the Americanstyle convertible bonds with the Heston stochastic volatility and that with the Cox–Ingersoll–Ross (CIR) stochastic interest rate are both considered. Due to the complexity of both problems, resulting from an additional stochastic factor, it is almost impossible to find any analytical solution. Therefore, a predictor–corrector scheme is chosen as the numerical scheme to solve the partial differential equations (PDEs), with the Douglas–Rachford (D–R) method being utilized as one of the Alternating Direction Implicit (ADI) methods for the correction step to obtain the numerical solution. Finally, the accuracy of our approach is numerically verified, and different properties of convertible bond price and the optimal conversion price are also demonstrated and discussed through examples.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Sha Lin, SongPing ZhuAbstractIn this paper, the pricing problem for the Americanstyle convertible bonds with the Heston stochastic volatility and that with the Cox–Ingersoll–Ross (CIR) stochastic interest rate are both considered. Due to the complexity of both problems, resulting from an additional stochastic factor, it is almost impossible to find any analytical solution. Therefore, a predictor–corrector scheme is chosen as the numerical scheme to solve the partial differential equations (PDEs), with the Douglas–Rachford (D–R) method being utilized as one of the Alternating Direction Implicit (ADI) methods for the correction step to obtain the numerical solution. Finally, the accuracy of our approach is numerically verified, and different properties of convertible bond price and the optimal conversion price are also demonstrated and discussed through examples.
 Meshless simulation of antiplane crack problems by the method of
fundamental solutions using the crack Green’s function Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Ji Ma, Wen Chen, Chuanzeng Zhang, Ji LinAbstractIn this paper, we present a meshless method of fundamental solutions using the analytical crack Green’s function to solve antiplane crack problems. The proposed scheme is a simple, powerful and effective collocation method for crack problems since it only requires the boundary discretization without special treatments of the crack. Three typical numerical examples, namely, a central crack, an offcentral crack and a central slant crack are presented and discussed to illustrate the accuracy, convergence and stability of the proposed method.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Ji Ma, Wen Chen, Chuanzeng Zhang, Ji LinAbstractIn this paper, we present a meshless method of fundamental solutions using the analytical crack Green’s function to solve antiplane crack problems. The proposed scheme is a simple, powerful and effective collocation method for crack problems since it only requires the boundary discretization without special treatments of the crack. Three typical numerical examples, namely, a central crack, an offcentral crack and a central slant crack are presented and discussed to illustrate the accuracy, convergence and stability of the proposed method.
 Predictor–Corrector Nodal Integral Method for simulation of high
Reynolds number fluid flow using larger time steps in Burgers ’ equation
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Niteen Kumar, Rudrodip Majumdar, Suneet SinghAbstractNodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient nonlinear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large timestep is to be considered. In order to resolve these difficulties, a new type of physicsbased predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobianfree NewtonKrylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a nonlinear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothonedimensional and twodimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Niteen Kumar, Rudrodip Majumdar, Suneet SinghAbstractNodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient nonlinear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large timestep is to be considered. In order to resolve these difficulties, a new type of physicsbased predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobianfree NewtonKrylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a nonlinear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothonedimensional and twodimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.

p biharmonic+system+with+negative+exponents&rft.title=Computers+&+Mathematics+with+Applications&rft.issn=08981221&rft.date=&rft.volume=">A critical p biharmonic system with negative exponents Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yanbin Sang, Yan RenAbstractIn this paper, we are devoted to the study of critical pbiharmonic system with a parameter λ, which involves stronglycoupled critical nonlinearities and negative exponents. We prove that there exists a positive constant λ∗ such that the above problem admits at least two solutions if λ∈(0,λ∗).
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yanbin Sang, Yan RenAbstractIn this paper, we are devoted to the study of critical pbiharmonic system with a parameter λ, which involves stronglycoupled critical nonlinearities and negative exponents. We prove that there exists a positive constant λ∗ such that the above problem admits at least two solutions if λ∈(0,λ∗).
 Fractional derivative modeling for axisymmetric consolidation of
multilayered crossanisotropic viscoelastic porous media Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Zhi Yong Ai, Yong Zhi Zhao, Wen Jie LiuAbstractThis paper presents a fractional derivative modeling for axisymmetric consolidation of multilayered crossanisotropic viscoelastic porous media. The elastic solution for multilayered porous media is acquired by the extended precise integration method. The flexibility coefficient of the fractional Merchant viscoelastic model is deduced in the Laplace transformed domain, and the viscoelastic solution of the problem is further obtained by the elastic–viscoelastic correspondence principle. The numerical results are compared with those of published literatures to verify the proposed method, and the influences of fractional derivative order, viscoelasticity of solid skeleton and layering of materials on the axisymmetric consolidation of multilayered viscoelastic porous media are investigated by several examples.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Zhi Yong Ai, Yong Zhi Zhao, Wen Jie LiuAbstractThis paper presents a fractional derivative modeling for axisymmetric consolidation of multilayered crossanisotropic viscoelastic porous media. The elastic solution for multilayered porous media is acquired by the extended precise integration method. The flexibility coefficient of the fractional Merchant viscoelastic model is deduced in the Laplace transformed domain, and the viscoelastic solution of the problem is further obtained by the elastic–viscoelastic correspondence principle. The numerical results are compared with those of published literatures to verify the proposed method, and the influences of fractional derivative order, viscoelasticity of solid skeleton and layering of materials on the axisymmetric consolidation of multilayered viscoelastic porous media are investigated by several examples.
 An efficient longtime stable secondorder accurate timestepping scheme
for evolutionary magnetomicropolar flows Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): S.S. RavindranAbstractWe propose and study a secondorder in time linearly extrapolated timestepping method, together with finite element spatial discretizations, for the coupled magnetomicropolar fluid model that describes the motion of relatively rigid microelements in electrically conducting flow in the presence of magnetic field. It is based on twostep backward difference time discretization and linear extrapolation of the coupling and nonlinear terms such that skew symmetry properties of the nonlinear terms are preserved. The algorithm requires one solve of linear magnetomicropolar system at each time step. We establish unconditional and long time stability of the scheme in the L2 norm, provided external data is uniformly bounded in time. Optimal order error estimates for fully discretized scheme using finite element spatial discretization are derived. Numerical examples are presented that illustrate the accuracy, efficiency and long time stability of the scheme.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): S.S. RavindranAbstractWe propose and study a secondorder in time linearly extrapolated timestepping method, together with finite element spatial discretizations, for the coupled magnetomicropolar fluid model that describes the motion of relatively rigid microelements in electrically conducting flow in the presence of magnetic field. It is based on twostep backward difference time discretization and linear extrapolation of the coupling and nonlinear terms such that skew symmetry properties of the nonlinear terms are preserved. The algorithm requires one solve of linear magnetomicropolar system at each time step. We establish unconditional and long time stability of the scheme in the L2 norm, provided external data is uniformly bounded in time. Optimal order error estimates for fully discretized scheme using finite element spatial discretization are derived. Numerical examples are presented that illustrate the accuracy, efficiency and long time stability of the scheme.
 Appropriate stabilized Galerkin approaches for solving twodimensional
coupled Burgers’ equations at high Reynolds numbers Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yong Chai, Jie OuyangAbstractThis paper aims to seek proper stabilized Galerkin methods for solving the twodimensional coupled Burgers’ equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov–Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank–Nicolson scheme for time discretization and the last one is applied in its semiimplicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convectiondominated problems.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Yong Chai, Jie OuyangAbstractThis paper aims to seek proper stabilized Galerkin methods for solving the twodimensional coupled Burgers’ equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov–Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank–Nicolson scheme for time discretization and the last one is applied in its semiimplicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convectiondominated problems.
 Fully meshless solution of the onedimensional multigroup neutron
transport equation with the radial basis function collocation method Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): T. Tanbay, B. OzgenerAbstractIn this paper a fully meshless method is proposed for the numerical solution of the onedimensional multigroup neutron transport equation with anisotropic scattering. Both firstorder and evenparity forms of the transport equation are studied. The radial basis function collocation method is chosen for the spatial treatment, and Legendre polynomials are used to approximate the angular variable. The selection of the Legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables. Multiquadric is utilized as the radial function. Seven problems are considered to evaluate the performance of the method. The results show that the method converges exponentially, and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains. For the onegroup isotropic benchmark problem, discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided, and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinatesfinite element scheme when the shape parameter is properly chosen.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): T. Tanbay, B. OzgenerAbstractIn this paper a fully meshless method is proposed for the numerical solution of the onedimensional multigroup neutron transport equation with anisotropic scattering. Both firstorder and evenparity forms of the transport equation are studied. The radial basis function collocation method is chosen for the spatial treatment, and Legendre polynomials are used to approximate the angular variable. The selection of the Legendre polynomials instead of discrete ordinates approach resulted with a fully meshless algorithm in both independent variables. Multiquadric is utilized as the radial function. Seven problems are considered to evaluate the performance of the method. The results show that the method converges exponentially, and it is possible to obtain high levels of accuracies for the multiplication factor and neutron flux with a good stability in both spatial and angular domains. For the onegroup isotropic benchmark problem, discrete ordinates solutions employing discontinuous linear finite elements for the spatial variable are also provided, and a comparison of the methods revealed that the fully meshless method produced more accurate results than the discrete ordinatesfinite element scheme when the shape parameter is properly chosen.
 Compact Integration Rules as a quadrature method with some applications
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Víctor J. Llorente, Antonio PascauAbstractIn many instances of computational science and engineering the value of a definite integral of a known function f(x) is required in an interval. Nowadays there are plenty of methods that provide this quantity with a given accuracy. In one way or another, all of them assume an interpolating function, usually polynomial, that represents the original function either locally or globally. This paper presents a new way of calculating ∫x1x2f(x)dx by means of compact integration, in a similar way to the compact differentiation employed in computational physics and mathematics. Compact integration is a linear combination of definite integrals associated to an interval and its adjacent ones, written in terms of nodal values of f(x). The coefficients that multiply both the integrals and f(x) at the nodes are obtained by matching terms in a Taylor series expansion. In this implicit method a system of algebraic equations is solved, where the vector of unknowns contains the integrals in each interval of a uniform discrete domain. As a result the definite integral over the whole domain is the sum of all these integrals. In this paper the mathematical tool is analyzed by deriving the appropriate coefficients for a given accuracy, and is exploited in various numerical examples and applications. The great accuracy of the method is highlighted.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): Víctor J. Llorente, Antonio PascauAbstractIn many instances of computational science and engineering the value of a definite integral of a known function f(x) is required in an interval. Nowadays there are plenty of methods that provide this quantity with a given accuracy. In one way or another, all of them assume an interpolating function, usually polynomial, that represents the original function either locally or globally. This paper presents a new way of calculating ∫x1x2f(x)dx by means of compact integration, in a similar way to the compact differentiation employed in computational physics and mathematics. Compact integration is a linear combination of definite integrals associated to an interval and its adjacent ones, written in terms of nodal values of f(x). The coefficients that multiply both the integrals and f(x) at the nodes are obtained by matching terms in a Taylor series expansion. In this implicit method a system of algebraic equations is solved, where the vector of unknowns contains the integrals in each interval of a uniform discrete domain. As a result the definite integral over the whole domain is the sum of all these integrals. In this paper the mathematical tool is analyzed by deriving the appropriate coefficients for a given accuracy, and is exploited in various numerical examples and applications. The great accuracy of the method is highlighted.
 Space–time residual distribution on moving meshes
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): M.E. Hubbard, M. Ricchiuto, D. SármányAbstractThis article investigates the potential for an radaptation algorithm to improve the efficiency of space–time residual distribution schemes in the approximation of timedependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually “relocated” as the flow evolves so that features of interest remain resolved as they move within the domain.Adaptive strategies of this type are well suited to the space–time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of timestep, even on the distorted space–time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the timestep, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixedmesh scheme is retained on the moving mesh in the majority of applications tested.Space–time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the socalled Cproperty (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved.The radaptation is applied within each timestep by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error.
 Abstract: Publication date: 1 March 2020Source: Computers & Mathematics with Applications, Volume 79, Issue 5Author(s): M.E. Hubbard, M. Ricchiuto, D. SármányAbstractThis article investigates the potential for an radaptation algorithm to improve the efficiency of space–time residual distribution schemes in the approximation of timedependent hyperbolic conservation laws, e.g. scalar advection, shallow water flows, on unstructured, triangular meshes. In this adaptive framework the connectivity of the mesh, and hence the number of degrees of freedom, remain fixed, but the mesh nodes are continually “relocated” as the flow evolves so that features of interest remain resolved as they move within the domain.Adaptive strategies of this type are well suited to the space–time residual distribution framework because, when the discrete representation is allowed to be discontinuous in time, these algorithms can be designed to be positive (and hence stable) for any choice of timestep, even on the distorted space–time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the timestep, which can instead be chosen according to accuracy requirements. The order of accuracy of the fixedmesh scheme is retained on the moving mesh in the majority of applications tested.Space–time schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law, so moving the mesh does not artificially change the flow volume for pure conservation laws. For shallow water flows over variable bed topography, the socalled Cproperty (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved.The radaptation is applied within each timestep by interleaving the iterations of the nonlinear solver with updates to mesh node positions. The node movement is driven by a monitor function based on weighted approximations of the scaled gradient and Laplacian of the local solution and regularised by a smoothing iteration. Numerical results are shown in two dimensions for both scalar advection and for shallow water flow over a variable bed which show that, even for this simple implementation of the mesh movement, reductions in cpu times of up to 60% can be attained without increasing the error.
 Comparison of two interfacial flow solvers: Specific case of a single
droplet impacting onto a deep pool Abstract: Publication date: Available online 6 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Shuang Wu, Jie Zhang, Qi Xiao, MingJiu NiAbstractA numerical study about a droplet impacting onto a deep pool is presented here by employing the Volume of Fluid method to track the interface. In this investigation, two open source solvers, named Gerris and Basilisk, are used to solve the incompressible Navier–Stokes equations with free surface. The results are compared to provide some reference for the researchers involved in this research community. Firstly, the capabilities of the two solvers in simulating the droplet impacting problems are validated against available experimental results. And the adaptive mesh refinement techniques used in the two solvers are also found to be applicable. When simulating the drop impacting problems in an axisymmetric coordinate system, it is found that less meshes are produced in Gerris than that in Basilisk when similar adaptive criterion is adopted in both solvers. However, Basilisk still shows much higher computational efficiency due to its great superiority at parallelization, and thus less CPU time is required for such unsteady problems. When simulating moderate Reynolds number impacting problems, the two numerical solvers show perfect agreement (Re≤5000), however, since higher Reynolds numbers (Re≥5000) require smaller size of mesh in vicinity of the thin impacting region and the concentrated vortex region, it is hard for Gerris to obtain satisfactory results because of its poor computational efficiency. Thus, with such a high parallel capability and computational efficiency, Basilisk is introduced to solve a large number of cases. It is summarized that the flow pattern of the drop impacting problem can be classified to three categories. They are smooth ejecta sheet, main vortex shedding and Von Kármán vortex street. Each flow pattern relates to the vortex evolution inside the liquid closely. Basililsk also shows its potential in solving much more complicated threedimensional problems. They cannot be completed by Gerris owing to its poor computational efficiency.
 Abstract: Publication date: Available online 6 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Shuang Wu, Jie Zhang, Qi Xiao, MingJiu NiAbstractA numerical study about a droplet impacting onto a deep pool is presented here by employing the Volume of Fluid method to track the interface. In this investigation, two open source solvers, named Gerris and Basilisk, are used to solve the incompressible Navier–Stokes equations with free surface. The results are compared to provide some reference for the researchers involved in this research community. Firstly, the capabilities of the two solvers in simulating the droplet impacting problems are validated against available experimental results. And the adaptive mesh refinement techniques used in the two solvers are also found to be applicable. When simulating the drop impacting problems in an axisymmetric coordinate system, it is found that less meshes are produced in Gerris than that in Basilisk when similar adaptive criterion is adopted in both solvers. However, Basilisk still shows much higher computational efficiency due to its great superiority at parallelization, and thus less CPU time is required for such unsteady problems. When simulating moderate Reynolds number impacting problems, the two numerical solvers show perfect agreement (Re≤5000), however, since higher Reynolds numbers (Re≥5000) require smaller size of mesh in vicinity of the thin impacting region and the concentrated vortex region, it is hard for Gerris to obtain satisfactory results because of its poor computational efficiency. Thus, with such a high parallel capability and computational efficiency, Basilisk is introduced to solve a large number of cases. It is summarized that the flow pattern of the drop impacting problem can be classified to three categories. They are smooth ejecta sheet, main vortex shedding and Von Kármán vortex street. Each flow pattern relates to the vortex evolution inside the liquid closely. Basililsk also shows its potential in solving much more complicated threedimensional problems. They cannot be completed by Gerris owing to its poor computational efficiency.
 Free vibration analysis of FGCNTRC shell structures using the meshfree
radial point interpolation method Abstract: Publication date: Available online 3 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): H. Mellouli, H. Jrad, M. Wali, F. DammakAbstractThis study conducts the firstknown free vibration analysis of functionally graded carbon nanotubesreinforced (FGCNTRC) shell structures using the meshfree radial point interpolation method (RPIM). The modified firstorder shear deformation theory (modified FSDT) is implemented to get the realistic effect of the transverse shear deformation with its parabolic distribution. Numerical examples are carried out to examine the convergence and accuracy of the elementfree RPIM method in its application to the free vibration FGCNTRC analysis of shell structures. Results obtained using the proposed meshfree method, demonstrate that the improved FSDT is very successful compared to closedform solutions and finite element results using different shell theories.
 Abstract: Publication date: Available online 3 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): H. Mellouli, H. Jrad, M. Wali, F. DammakAbstractThis study conducts the firstknown free vibration analysis of functionally graded carbon nanotubesreinforced (FGCNTRC) shell structures using the meshfree radial point interpolation method (RPIM). The modified firstorder shear deformation theory (modified FSDT) is implemented to get the realistic effect of the transverse shear deformation with its parabolic distribution. Numerical examples are carried out to examine the convergence and accuracy of the elementfree RPIM method in its application to the free vibration FGCNTRC analysis of shell structures. Results obtained using the proposed meshfree method, demonstrate that the improved FSDT is very successful compared to closedform solutions and finite element results using different shell theories.
 A twostage adaptive scheme based on RBF collocation for solving elliptic
PDEs Abstract: Publication date: Available online 2 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): R. Cavoretto, A. De RossiAbstractIn this paper we present a new adaptive twostage algorithm for solving elliptic partial differential equations via a radial basis function collocation method. Our adaptive meshless scheme is based at first on the use of a leaveoneout cross validation technique, and then on a residual subsampling method. Each of phases is characterized by different error indicators and refinement strategies. The combination of these computational approaches allows us to detect the areas that need to be refined, also including the chance to further add or remove adaptively any points. The resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures. Several numerical experiments support our study by illustrating the performance of our twostage scheme. Finally, the latter is also compared with an efficient adaptive finite element method.
 Abstract: Publication date: Available online 2 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): R. Cavoretto, A. De RossiAbstractIn this paper we present a new adaptive twostage algorithm for solving elliptic partial differential equations via a radial basis function collocation method. Our adaptive meshless scheme is based at first on the use of a leaveoneout cross validation technique, and then on a residual subsampling method. Each of phases is characterized by different error indicators and refinement strategies. The combination of these computational approaches allows us to detect the areas that need to be refined, also including the chance to further add or remove adaptively any points. The resulting algorithm turns out to be flexible and effective through a good interaction between error indicators and refinement procedures. Several numerical experiments support our study by illustrating the performance of our twostage scheme. Finally, the latter is also compared with an efficient adaptive finite element method.
 The DPGstar method
 Abstract: Publication date: Available online 1 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Leszek Demkowicz, Jay Gopalakrishnan, Brendan KeithAbstractThis article introduces the DPGstar (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov–Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddlepoint problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG* and DPG methods can be seen as generalizations of LL∗ and leastsquares methods, respectively. A priori error analysis and a posteriori error control for the DPG* method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG* and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.
 Abstract: Publication date: Available online 1 February 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Leszek Demkowicz, Jay Gopalakrishnan, Brendan KeithAbstractThis article introduces the DPGstar (from now on, denoted DPG*) finite element method. It is a method that is in some sense dual to the discontinuous Petrov–Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG* methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddlepoint problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG* and DPG methods can be seen as generalizations of LL∗ and leastsquares methods, respectively. A priori error analysis and a posteriori error control for the DPG* method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG* and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable.
 Effect of thermal radiation on conjugate natural convection flow of a
micropolar fluid along a vertical surface Abstract: Publication date: Available online 28 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Sadia Siddiqa, Naheed Begum, Md. Anwar Hossain, M.N. Abrar, Rama Subba Reddy Gorla, Qasem AlMdallalAbstractThis article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the nonsimilarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Kellerbox method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K, conjugate parameter, B, and thermal radiation parameter, Rd, are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the microinertia parameter increases. All the physical quantities get augmented with thermal radiation.
 Abstract: Publication date: Available online 28 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Sadia Siddiqa, Naheed Begum, Md. Anwar Hossain, M.N. Abrar, Rama Subba Reddy Gorla, Qasem AlMdallalAbstractThis article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the nonsimilarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Kellerbox method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K, conjugate parameter, B, and thermal radiation parameter, Rd, are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the microinertia parameter increases. All the physical quantities get augmented with thermal radiation.
 Multigoaloriented optimal control problems with nonlinear PDE constraints
 Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): B. Endtmayer, U. Langer, I. Neitzel, T. Wick, W. WollnerAbstractIn this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semilinear monotone PDE and the regularized pLaplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dualweighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated with the help of several numerical examples.
 Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): B. Endtmayer, U. Langer, I. Neitzel, T. Wick, W. WollnerAbstractIn this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semilinear monotone PDE and the regularized pLaplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dualweighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated with the help of several numerical examples.
 Numerical analysis of a contact problem with wear
 Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Danfu Han, Weimin Han, Michal Jureczka, Anna OchalAbstractThis paper represents a sequel to Jureczka and Ochal (2019) where numerical solution of a quasistatic contact problem is considered for an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. Some preliminary error analysis for a fully discrete approximation of the contact problem was provided in Jureczka and Ochal (2019). In this paper, we consider a more general fully discrete numerical scheme for the contact problem, derive optimal order error bounds and present computer simulation results showing that the numerical convergence orders match the theoretical predictions.
 Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Danfu Han, Weimin Han, Michal Jureczka, Anna OchalAbstractThis paper represents a sequel to Jureczka and Ochal (2019) where numerical solution of a quasistatic contact problem is considered for an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. Some preliminary error analysis for a fully discrete approximation of the contact problem was provided in Jureczka and Ochal (2019). In this paper, we consider a more general fully discrete numerical scheme for the contact problem, derive optimal order error bounds and present computer simulation results showing that the numerical convergence orders match the theoretical predictions.
 waLBerla:+A+blockstructured+highperformance+framework+for+multiphysics+simulations&rft.title=Computers+&+Mathematics+with+Applications&rft.issn=08981221&rft.date=&rft.volume=">waLBerla: A blockstructured highperformance framework for multiphysics
simulations Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Martin Bauer, Sebastian Eibl, Christian Godenschwager, Nils Kohl, Michael Kuron, Christoph Rettinger, Florian Schornbaum, Christoph Schwarzmeier, Dominik Thönnes, Harald Köstler, Ulrich RüdeAbstractProgramming current supercomputers efficiently is a challenging task. Multiple levels of parallelism on the core, on the compute node, and between nodes need to be exploited to make full use of the system. Heterogeneous hardware architectures with accelerators further complicate the development process. waLBerla addresses these challenges by providing the user with highly efficient building blocks for developing simulations on blockstructured grids. The blockstructured domain partitioning is flexible enough to handle complex geometries, while the structured grid within each block allows for highly efficient implementations of stencilbased algorithms. We present several example applications realized with waLBerla, ranging from lattice Boltzmann methods to rigid particle simulations. Most importantly, these methods can be coupled together, enabling multiphysics simulations. The framework uses metaprogramming techniques to generate highly efficient code for CPUs and GPUs from a symbolic method formulation.
 Abstract: Publication date: Available online 25 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Martin Bauer, Sebastian Eibl, Christian Godenschwager, Nils Kohl, Michael Kuron, Christoph Rettinger, Florian Schornbaum, Christoph Schwarzmeier, Dominik Thönnes, Harald Köstler, Ulrich RüdeAbstractProgramming current supercomputers efficiently is a challenging task. Multiple levels of parallelism on the core, on the compute node, and between nodes need to be exploited to make full use of the system. Heterogeneous hardware architectures with accelerators further complicate the development process. waLBerla addresses these challenges by providing the user with highly efficient building blocks for developing simulations on blockstructured grids. The blockstructured domain partitioning is flexible enough to handle complex geometries, while the structured grid within each block allows for highly efficient implementations of stencilbased algorithms. We present several example applications realized with waLBerla, ranging from lattice Boltzmann methods to rigid particle simulations. Most importantly, these methods can be coupled together, enabling multiphysics simulations. The framework uses metaprogramming techniques to generate highly efficient code for CPUs and GPUs from a symbolic method formulation.
 A class of generalized mixed variational–hemivariational inequalities I:
Existence and uniqueness results Abstract: Publication date: Available online 24 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yunru Bai, Stanisław Migórski, Shengda ZengAbstractWe investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational–hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan–Knaster–Kuratowski–Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.
 Abstract: Publication date: Available online 24 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Yunru Bai, Stanisław Migórski, Shengda ZengAbstractWe investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational–hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan–Knaster–Kuratowski–Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.
 Tent pitching and TrefftzDG method for the acoustic wave equation
 Abstract: Publication date: Available online 24 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ilaria Perugia, Joachim Schöberl, Paul Stocker, Christoph WintersteigerAbstractWe present a space–time Trefftz discontinuous Galerkin method for approximating the acoustic wave equation semiexplicitly on tent pitched meshes. DG Trefftz methods use discontinuous test and trial functions, which solve the wave equation locally. Tent pitched meshes allow us to solve the equation elementwise, allowing locally optimal advances in time. The method is implemented in NGSolve, solving the space–time elements in parallel, whenever possible. Insights into the implementational details are given, including the case of propagation in heterogeneous media.
 Abstract: Publication date: Available online 24 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ilaria Perugia, Joachim Schöberl, Paul Stocker, Christoph WintersteigerAbstractWe present a space–time Trefftz discontinuous Galerkin method for approximating the acoustic wave equation semiexplicitly on tent pitched meshes. DG Trefftz methods use discontinuous test and trial functions, which solve the wave equation locally. Tent pitched meshes allow us to solve the equation elementwise, allowing locally optimal advances in time. The method is implemented in NGSolve, solving the space–time elements in parallel, whenever possible. Insights into the implementational details are given, including the case of propagation in heterogeneous media.
 Poincaré–Friedrichs type constants for operators involving grad, curl,
and div: Theory and numerical experiments Abstract: Publication date: Available online 23 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Dirk Pauly, Jan ValdmanAbstractWe give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants cn>0 arising in estimates of the form u L2(Ω)≤c0 gradu L2(Ω), E L2(Ω)≤c1 curlE L2(Ω), H L2(Ω)≤c2 divH L2(Ω).Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.
 Abstract: Publication date: Available online 23 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Dirk Pauly, Jan ValdmanAbstractWe give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants cn>0 arising in estimates of the form u L2(Ω)≤c0 gradu L2(Ω), E L2(Ω)≤c1 curlE L2(Ω), H L2(Ω)≤c2 divH L2(Ω).Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.
 Temporal second order difference schemes for the multidimensional
variableorder time fractional subdiffusion equations Abstract: Publication date: Available online 23 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ruilian Du, Anatoly A. Alikhanov, ZhiZhong SunAbstractA special point on each time interval is found for the approximation of the variableorder time Caputo derivative, which makes at least second order approximation accuracy be obtained. On this basis, two difference schemes are proposed for the multidimensional variableorder time fractional subdiffusion equations, which have second order accuracy in time, second order and fourth order accuracy in space, respectively. The obtained difference schemes are proved to be uniquely solvable. The convergence and stability of the schemes in the discrete H1norm are analyzed by utilizing the energy method. Some numerical examples are presented to verify the theoretical results.
 Abstract: Publication date: Available online 23 January 2020Source: Computers & Mathematics with ApplicationsAuthor(s): Ruilian Du, Anatoly A. Alikhanov, ZhiZhong SunAbstractA special point on each time interval is found for the approximation of the variableorder time Caputo derivative, which makes at least second order approximation accuracy be obtained. On this basis, two difference schemes are proposed for the multidimensional variableorder time fractional subdiffusion equations, which have second order accuracy in time, second order and fourth order accuracy in space, respectively. The obtained difference schemes are proved to be uniquely solvable. The convergence and stability of the schemes in the discrete H1norm are analyzed by utilizing the energy method. Some numerical examples are presented to verify the theoretical results.