Authors:Guanyu Zhou Pages: 1 - 21 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Guanyu Zhou We consider the fictitious domain method with H 1 -penalty for the Stokes problem with Dirichlet boundary condition. First, for the continuous penalty problem, we obtain the optimal error estimate O ( ϵ ) for both the velocity and pressure, where ϵ is the penalty parameter. Moreover, we investigate the H m -regularity for the solution of the penalty problem. Then, we apply the finite element method with the P1/P1 element to the penalty problem. Since the solution to the penalty problem has a jump in the traction vector, we introduce some interpolation/projection operators, as well as an inf-sup condition with the norm depending on ϵ. With the help of these preliminaries, we derive the error estimates for the finite element approximation. The theoretical results are verified by the numerical experiments.

Authors:R. Donat; F. Guerrero; P. Mulet Pages: 22 - 42 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): R. Donat, F. Guerrero, P. Mulet Chromatographic processes can be modeled by nonlinear, convection-dominated partial differential equations, together with nonlinear relations: the adsorption isotherms. In this paper we consider the nonlinear equilibrium dispersive (ED) model with adsorption isotherms of Langmuir type. We show that, in this case, the ED model can be written as a system of conservation laws when the dispersion coefficient vanishes. We also show that the function that relates the conserved variables and the physically observed concentrations of the components in the mixture is one to one and it admits a global inverse, which cannot be given explicitly, but can be adequately computed. As a result, fully conservative numerical schemes can be designed for the ED model in chromatography. We propose a Weighted-Essentially-non-Oscillatory second order IMEX scheme and describe the numerical issues involved in its application. Through a series of numerical experiments, we show that our scheme gives accurate numerical solutions which capture the sharp discontinuities present in the chromatographic fronts, with the same stability restrictions as in the purely hyperbolic case.

Authors:Stefano Cipolla; Fabio Durastante Pages: 43 - 57 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Stefano Cipolla, Fabio Durastante In this paper, using an optimize-then-discretize approach, we address the numerical solution of two Fraction Partial Differential Equation constrained optimization problems: the Fractional Advection Dispersion Equation (FADE) and the two-dimensional semilinear Riesz Space Fractional Diffusion equation. Both a theoretical and experimental analysis of the problem is carried out. The algorithmic framework is based on the L-BFGS method coupled with a Krylov subspace solver. A suitable preconditioning strategy by approximate inverses is taken into account. Graphics Processing Unit (GPU) accelerator is used in the construction of the preconditioners. The numerical experiments are performed with benchmarked software/libraries enforcing the reproducibility of the results.

Authors:Eduardo M. Garau; Rafael Vázquez Pages: 58 - 87 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Eduardo M. Garau, Rafael Vázquez In this article we introduce all the ingredients to develop adaptive isogeometric methods based on hierarchical B-splines. In particular, we give precise definitions of local refinement and coarsening that, unlike previously existing methods, can be understood as the inverse of each other. We also define simple and intuitive data structures for the implementation of hierarchical B-splines, and algorithms for refinement and coarsening that take advantage of local information. We complete the paper with some simple numerical tests to show the performance of the data structures and algorithms, that have been implemented in the open-source Octave/Matlab code GeoPDEs.

Authors:Jin Zhang; Xiaowei Liu Pages: 88 - 98 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Jin Zhang, Xiaowei Liu A continuous interior penalty method with piecewise polynomials of degree p ≥ 2 is applied on a Shishkin mesh to solve a singularly perturbed convection–diffusion problem, whose solution has a single boundary layer. This method is analyzed by means of a series of integral identities developed for the convection terms. Then we prove a supercloseness bound of order 5/2 for a vertex-cell interpolation when p = 2 . The sharpness of our analysis is supported by some numerical experiments. Moreover, numerical tests show supercloseness clearly for p ≥ 3 .

Authors:Farhad Fakhar-Izadi; Mehdi Dehghan Pages: 99 - 120 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Farhad Fakhar-Izadi, Mehdi Dehghan The numerical approximation of solution to nonlinear parabolic Volterra and Fredholm partial integro-differential equations is studied in this paper. Unlike the conventional methods which discretize the time variable by finite difference schemes, we use the spectral method for this purpose. Indeed, both of the space and time discretizations are based on the Legendre-collocation method which lead to conversion of the problem to a nonlinear system of algebraic equations. The convergence of the proposed method is proven by providing an L ∞ error estimate. Several numerical examples are included to demonstrate the efficiency and spectral accuracy of the proposed method in the space and time directions.

Authors:Pouria Assari; Mehdi Dehghan Pages: 137 - 158 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Pouria Assari, Mehdi Dehghan The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

Authors:S. Kopecz; A. Meister Pages: 159 - 179 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): S. Kopecz, A. Meister In [6] the modified Patankar–Euler and modified Patankar–Runge–Kutta schemes were introduced to solve positive and conservative systems of ordinary differential equations. These modifications of the forward Euler scheme and Heun's method guarantee positivity and conservation irrespective of the chosen time step size. In this paper we introduce a general definition of modified Patankar–Runge–Kutta schemes and derive necessary and sufficient conditions to obtain first and second order methods. We also introduce two novel families of two-stage second order modified Patankar–Runge–Kutta schemes.

Authors:Chenguang Zhou; Yongkui Zou; Shimin Chai; Qian Zhang; Hongze Zhu Pages: 180 - 199 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Chenguang Zhou, Yongkui Zou, Shimin Chai, Qian Zhang, Hongze Zhu In this paper, we apply a new weak Galerkin mixed finite element method (WGMFEM) with stabilization term to solve heat equations. This method allows the usage of totally discontinuous functions in the approximation space. The WGMFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. In addition, we develop and analyze the error estimates for both continuous and discontinuous time WGMFEM schemes. Optimal order error estimates in both L 2 and triple-bar ⫼ ⋅ ⫼ norms are established, respectively. Finally, numerical tests are conducted to illustrate the theoretical results.

Authors:Hanyu Li; Shaoxin Wang Pages: 200 - 220 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Hanyu Li, Shaoxin Wang The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, we first present an expression of the partial unified condition number when the data space is measured by a general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.

Authors:Arijit Hazra; Gert Lube; Hans-Georg Raumer Pages: 241 - 255 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Arijit Hazra, Gert Lube, Hans-Georg Raumer Magnetic Resonance Imaging (MRI) is a widely applied non-invasive imaging modality based on non-ionizing radiation which gives excellent images and soft tissue contrast of living tissues. We consider the modified Bloch problem as a model of MRI for flowing spins in an incompressible flow field. After establishing the well-posedness of the corresponding evolution problem, we analyze its spatial semi-discretization using discontinuous Galerkin methods. The high frequency time evolution requires a proper explicit and adaptive temporal discretization. The applicability of the approach is shown for basic examples.

Authors:María González; Virginia Selgas Pages: 275 - 299 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): María González, Virginia Selgas We consider a fluid–structure interaction problem consisting of the time-dependent Stokes equations in the fluid domain coupled with the equations of linear elastodynamics in the solid domain. For simplicity, all changes of geometry are neglected. We propose a new method in terms of the fluid velocity, the fluid pressure, the structural velocity and the Cauchy stress tensor. We show that the new weak formulation is well-posed. Then, we propose a new semidiscrete problem where the velocities and the fluid pressure are approximated using a stable pair for the Stokes problem in the fluid domain and compatible finite elements in the solid domain. We obtain a priori estimates for the solution of the semidiscrete problem, prove the convergence of these solutions to the solution of the weak formulation and obtain error estimates. A time discretization based on the backward Euler method leads to a fully discrete scheme in which the computation of the approximated Cauchy stress tensor can be decoupled from that of the remaining unknowns at each time step. The displacements in the structure (if needed) can be recovered by quadrature. Finally, some numerical experiments showing the performance of the method are provided.

Authors:Francesco Mezzadri; Emanuele Galligani Pages: 300 - 319 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Francesco Mezzadri, Emanuele Galligani In this paper we solve a 2D nonlinear, non-steady reaction–convection–diffusion equation subject to Dirichlet boundary conditions by an iterative procedure consisting in lagging the diffusion term. First, we analyze the procedure, which we call Lagged Diffusivity Method. In particular, we provide a proof of the uniqueness of the solution and of the convergence of the lagged iteration when some assumptions are satisfied. We also describe outer and inner solvers, with special regard to how to link the stopping criteria in an efficient way. Numerical experiments are then introduced in order to evaluate the role of different linear solvers and of other components of the solution procedure, considering also the effect of the discretization.

Authors:Mahboub Baccouch Pages: 18 - 37 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Mahboub Baccouch In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L 2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the ( p + 1 ) -degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L 2 -norm with order of convergence p + 2 . Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p + 2 in the L 2 -norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for P p polynomials with p ≥ 1 . Several numerical results are presented to validate the theoretical results.

Authors:Andrea Barth; Franz G. Fuchs Pages: 38 - 51 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Andrea Barth, Franz G. Fuchs In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein–Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings.

Authors:Lars Grüne; Thuy T.T. Le Pages: 68 - 81 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Lars Grüne, Thuy T.T. Le We introduce a new formulation of the minimum time problem in which we employ the signed minimum time function positive outside of the target, negative in its interior and zero on its boundary. Under some standard assumptions, we prove the so called Bridge Dynamic Programming Principle (BDPP) which is a relation between the value functions defined on the complement of the target and in its interior. Then owing to BDPP, we obtain the error estimates of a semi-Lagrangian discretization of the resulting Hamilton–Jacobi–Bellman equation. In the end, we provide numerical tests and error comparisons which show that the new approach can lead to significantly reduced numerical errors.

Authors:Fanhai Zeng; Changpin Li Pages: 82 - 95 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Fanhai Zeng, Changpin Li In this paper, a new Crank–Nicolson finite element method for the time-fractional subdiffusion equation is developed, in which a novel time discretization called the modified L1 method is used to discretize the Riemann–Liouville fractional derivative. The present method is unconditionally stable and convergent of order O ( τ 1 + β + h r + 1 ) , where β ∈ ( 0 , 1 ) , τ and h are the step sizes in time and space, respectively, and r is the degree of the piecewise polynomial space. The derived method is reduced to the classical Crank–Nicolson method when β → 1 . The new time discretization is also used to solve the fractional cable equation. And the unconditional stability and convergence are given. Numerical examples are provided which support the theoretical analysis. The comparison with the existing methods are also given, which shows good performances of the present methods.

Authors:Armin Westerkamp; Manuel Torrilhon Pages: 96 - 114 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Armin Westerkamp, Manuel Torrilhon We present an analysis of a set of parametrized boundary conditions for a Stokes–Brinkman model in two space dimensions, discretized by finite elements. We particularly point out an instability which arises when these boundary conditions are posed on a curved line, which then leads to unphysical oscillations. In contrast to a Navier-slip condition, which is prone to Babuška's paradox, this instability can be traced back to the continuous level. We claim that the stability in these cases depend on the amount of curvature at the boundary, which is shown in a reduced setting in cylinder coordinates. The transition to a two dimensional Cartesian case is then based on numerical studies, which further substantiate the claim. Lastly, stabilization techniques are motivated that enhance the continuous FEM setting and are conveniently able to deal with arising oscillations.

Authors:Asma Toumi; Guillaume Dufour; Ronan Perrussel; Thomas Unfer Pages: 115 - 133 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Asma Toumi, Guillaume Dufour, Ronan Perrussel, Thomas Unfer We propose an asynchronous method for the explicit integration of multi-scale partial differential equations. This method is restricted by a local CFL (Courant Friedrichs Lewy) condition rather than the traditional global CFL condition. Moreover, contrary to other local time-stepping (LTS) methods, the asynchronous algorithm permits the selection of independent time steps in each mesh element. We derived an asynchronous Runge–Kutta 2 (ARK2) scheme from a standard explicit Runge–Kutta method and we proved that the ARK2 scheme is second order convergent. Comparing with the classical integration, the asynchronous scheme is effective in terms of computation time.

Authors:Ömür Kıvanç Kürkçü; Ersin Aslan; Mehmet Sezer Pages: 134 - 148 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Ömür Kıvanç Kürkçü, Ersin Aslan, Mehmet Sezer In this study, we solve some widely-used model problems consisting of linear, nonlinear differential and integral equations, employing Dickson polynomials with the parameter-α and the collocation points for an efficient matrix method. The convergence of a Dickson polynomial solution of the model problem is investigated by means of the residual function. We encode useful computer programs for model problems, in order to obtain the precise Dickson polynomial solutions. These solutions are plotted along with the exact solutions in figures and the numerical results are compared with other well-known methods in tables.

Authors:A.I. Ávila; A. Meister; M. Steigemann Pages: 149 - 169 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): A.I. Ávila, A. Meister, M. Steigemann Nonlinear time-dependent Schrödinger equations (NLSE) model several important problems in quantum physics and morphogenesis. Recently, vortex lattice formation were experimentally found in Bose–Einstein condensate and Fermi superfluids, which are modeled by adding a rotational term in the NLSE equation. Numerical solutions have been computed by using separate approaches for time and space variables. If we see the complex equation as a system, wave methods can be used. In this article, we consider finite element approximations using continuous Galerkin schemes in time and space by adaptive mesh balancing both errors. To get this balance, we adapt the dual weighted residual method used for wave equations and estimates of error indicators for adaptive space–time finite element discretization. The results show how important is dynamic refinement to control the degrees of freedom in space.

Authors:Peng Zhu; Shenglan Xie; Xiaoshen Wang Pages: 170 - 184 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Peng Zhu, Shenglan Xie, Xiaoshen Wang In this paper, the fractional cable equation, involving two Riemann–Liouville fractional derivatives, with initial/boundary condition is considered. Two fully discrete schemes are obtained by employing piecewise linear Galerkin FEM in space, and using convolution quadrature methods based on the first- and second-order backward difference methods in time. Optimal error estimates in terms of the initial data and the inhomogeneity for the semi-discrete scheme and fully discrete schemes are discussed. Numerical results are shown to verify the theoretical results.

Authors:Xuenan Sun; Xuezhang Liang Pages: 185 - 197 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Xuenan Sun, Xuezhang Liang A new algorithm for the reconstruction of two-dimensional (2D) images from projections is given. The algorithm is based on the expansions of Hakopian interpolation polynomial into Chebyshev–Fourier series. The computer simulation experiments show that the new algorithm is effective.

Authors:Jianchao Bai; Jicheng Li; Pingfan Dai Pages: 223 - 233 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Jianchao Bai, Jicheng Li, Pingfan Dai This work is devoted to designing a unified alternating update method for solving a class of structured low rank approximations under the convex and unitarily invariant norm. By the aid of the variational inequality and monotone operator, the proposed method is proved to converge to the solution point of an equivalent variational inequality with a worst-case O ( 1 / t ) convergence rate in a nonergodic sense. We also analyze that the involved subproblems under the Frobenius norm are respectively equivalent to the structured least-squares problem and low rank least-squares problem, where the explicit solutions to some special cases are derived. In order to investigate the efficiency of the proposed method, several examples in system identification are tested to validate that the proposed method can outperform some state-of-the-art methods.

Authors:Z. Jackiewicz; H. Mittelmann Pages: 234 - 248 Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Z. Jackiewicz, H. Mittelmann For many systems of differential equations modeling problems in science and engineering, there are often natural splittings of the right hand side into two parts, one of which is non-stiff or mildly stiff, and the other part is stiff. Such systems can be efficiently treated by a class of implicit–explicit (IMEX) diagonally implicit multistage integration methods (DIMSIMs), where the stiff part is integrated by implicit formula, and the non-stiff part is integrated by an explicit formula. We analyze stability of these methods when the implicit and explicit parts interact with each other. We look for methods with large absolute stability region, assuming that the implicit part of the method is A ( α ) -, A-, or L-stable. Finally, we furnish examples of IMEX DIMSIMs of order p = 5 and p = 6 and stage order q = p , with good stability properties. Numerical examples illustrate that the IMEX schemes constructed in this paper do not suffer from order reduction phenomenon for some range of stepsizes.

Authors:Kazuho Ito Pages: 1 - 20 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Kazuho Ito An energy conserving spectral scheme is presented for approximating the smooth solution of the dynamic elastica with free ends. The spatial discretization of the elastica is done on the basis of Galerkin spectral methods with a Legendre grid. It is established that the scheme has the unique solution and enjoys a spectral accuracy with respect to the size of the spatial grid. Moreover, some results of a numerical simulation are given to verify that the implemented scheme preserves the discrete energy.

Authors:Mariantonia Cotronei; Nada Sissouno Pages: 21 - 34 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Mariantonia Cotronei, Nada Sissouno The aim of the paper is to present Hermite-type multiwavelets, i.e. wavelets acting on vector data representing function values and consecutive derivatives, which satisfy the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. Such functions satisfy a two-scale relation which is level-dependent as well as the corresponding multiresolution analysis. An important feature of the associated filters is the possibility of factorizing their symbols in terms of the so-called cancellation operator. This is shown, in particular, in the situation where Hermite multiwavelets are obtained by completing interpolatory level-dependent Hermite subdivision operators, reproducing polynomial and exponential data, to biorthogonal systems. A few constructions of families of multiwavelet filters of this kind are proposed.

Authors:Peyman Hessari; Byeong-Chun Shin Pages: 35 - 52 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Peyman Hessari, Byeong-Chun Shin The subject of this paper is to investigate the first order system least squares Legendre and Chebyshev pseudo-spectral methods for coupled Stokes–Darcy equations. By introducing strain tensor as a new variable, Stokes–Darcy equations recast into a system of first order differential equations. The least squares functional is defined by summing up the weighted L 2 -norm of residuals of the first order system for coupled Stokes–Darcy equations. To treat Beavers–Joseph–Saffman interface conditions, the weighted L 2 -norm of these conditions are also added to the least squares functional. Continuous and discrete homogeneous functionals are shown to be equivalent to the combination of weighted H ( div ) and H 1 -norm for Stokes–Darcy equations. The spectral convergence for the Legendre and Chebyshev methods are derived. To demonstrate this analysis, numerical experiments are also presented.

Authors:Yuan-Ming Wang Pages: 53 - 67 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Yuan-Ming Wang This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L 2 , H 1 and L ∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L 2 , H 1 and L ∞ norms.

Authors:Jialin Hong; Lihai Ji; Linghua Kong; Tingchun Wang Pages: 68 - 81 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Jialin Hong, Lihai Ji, Linghua Kong, Tingchun Wang It has been pointed out in literature that the symplectic scheme of a nonlinear Hamiltonian system can not preserve the total energy in the discrete sense Ge and Marsden (1988) [10]. Moreover, due to the difficulty in obtaining a priori estimate of the numerical solution, it is very hard to establish the optimal error bound of the symplectic scheme without any restrictions on the grid ratios. In this paper, we develop and analyze a compact scheme for solving nonlinear Schrödinger equation. We introduce a cut-off technique for proving optimal L ∞ error estimate for the compact scheme. We show that the convergence of the compact scheme is of second order in time and of fourth order in space. Meanwhile, we define a new type of energy functional by using a recursion relationship, and then prove that the compact scheme is mass and energy-conserved, symplectic-conserved, unconditionally stable and can be computed efficiently. Numerical experiments confirm well the theoretical analysis results.

Authors:Xin-He Miao; Jian-Tao Yang; B. Saheya; Jein-Shan Chen Pages: 82 - 96 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Xin-He Miao, Jian-Tao Yang, B. Saheya, Jein-Shan Chen In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

Authors:Hailong Qiu; Rong An; Liquan Mei; Changfeng Xue Pages: 97 - 114 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Hailong Qiu, Rong An, Liquan Mei, Changfeng Xue Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions are considered in this paper. Our algorithms consist of solving one Navier–Stokes variational inequality problem used the linear equal-order finite element pair (i.e., P 1 – P 1 ) and then solving a linearization variational inequality problem used the quadratic equal-order finite element pair (i.e., P 2 – P 2 ). Moreover, the stability and convergence of our two-step algorithms are derived. Finally, numerical tests are presented to check theoretical results.

Authors:Ivan Sofronov Pages: 115 - 124 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Ivan Sofronov In [22] we announced equations for yielding differential operators of transparent boundary conditions (TBCs) for a certain class of second order hyperbolic systems. Here we present the full derivation of these equations and consider ways of their solving. The solutions represent local parts of TBCs, and they can be used as approximate nonreflecting boundary conditions. We give examples of computing such conditions called ‘truncated TBCs’ for 3D elasticity and Biot poroelasticity

Authors:J.A. Ferreira; D. Jordão; L. Pinto Pages: 125 - 140 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): J.A. Ferreira, D. Jordão, L. Pinto In this paper we propose a numerical scheme for wave type equations with damping and space variable coefficients. Relevant equations of this kind arise for instance in the context of Maxwell's equations, namely, the electric potential equation and the electric field equation. The main motivation to study such class of equations is the crucial role played by the electric potential or the electric field in enhanced drug delivery applications. Our numerical method is based on piecewise linear finite element approximation and it can be regarded as a finite difference method based on non-uniform partitions of the spatial domain. We show that the proposed method leads to second order convergence, in time and space, for the kinetic and potential energies with respect to a discrete L 2 -norm.

Authors:Thinh Kieu Pages: 141 - 164 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Thinh Kieu In this paper, we consider the generalized Forchheimer flows for slightly compressible fluids in porous media. Using Muskat's and Ward's general form of Forchheimer equations, we describe the flow of a single-phase fluid in R d , d ≥ 2 by a nonlinear degenerate system of density and momentum. A mixed finite element method is proposed for the approximation of the solution of the above system. The stability of the approximations are proved; the error estimates are derived for the numerical approximations for both continuous and discrete time procedures. The continuous dependence of numerical solutions on physical parameters are demonstrated. Experimental studies are presented regarding convergence rates and showing the dependence of the solution on the physical parameters.

Authors:G. Califano; G. Izzo; Z. Jackiewicz Pages: 165 - 175 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): G. Califano, G. Izzo, Z. Jackiewicz We present a systematic approach to the construction of starting procedures for general linear methods (GLMs) of order p and stage order q = p . Order conditions for starting procedures based on the generalized Runge–Kutta (RK) are derived using the theory of rooted trees, elementary differentials, and elementary weights, and examples of generalized RK formulas are given up to the order p = 4 .

Authors:Aleksandr E. Kolesov; Michael V. Klibanov; Loc H. Nguyen; Dinh-Liem Nguyen; Nguyen T. Thành Pages: 176 - 196 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Aleksandr E. Kolesov, Michael V. Klibanov, Loc H. Nguyen, Dinh-Liem Nguyen, Nguyen T. Thành The recently developed globally convergent numerical method for an inverse medium problem with the data resulting from a single measurement, proposed in [23], is tested on experimental data. The data were originally collected in the time domain, whereas the method works in the frequency domain with the multi-frequency data. Due to a significant amount of noise in the measured data, a straightforward application of the Fourier transform to these data does not work. Hence, we develop a heuristic data preprocessing procedure, which is described in the paper. The preprocessed data are used as the input for the inversion algorithm. Numerical results demonstrate a good accuracy of the reconstruction of both refractive indices and locations of targets.

Authors:Hossein Beyrami; Taher Lotfi; Katayoun Mahdiani Pages: 197 - 214 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Hossein Beyrami, Taher Lotfi, Katayoun Mahdiani In this article, we approximate the solution of the weakly singular Volterra integral equation of the second kind using the reproducing kernel Hilbert space (RKHS) method. This method does not require any background mesh and can easily be implemented. Since the solution of the second kind weakly singular Volterra integral equation has unbounded derivative at the left end point of the interval of the integral equation domain, RKHS method has poor convergence rate on the conventional uniform mesh. Consequently, the graded mesh is proposed. Using error analysis, we show the RKHS method has better convergence rate on the graded mesh than the uniform mesh. Numerical examples are given to confirm the error analysis results. Regularization of the solution is an alternative approach to improve the efficiency of the RKHS method. In this regard, an smooth transformation is used to regularization and obtained numerical results are compared with other methods.

Abstract: Publication date: Available online 12 October 2017 Source:Applied Numerical Mathematics Author(s): Qili Tang, Yunqing Huang A fully discrete Crank-Nicolson leap-frog (CNLF) scheme is presented and studied for the nonstationary incompressible Navier-Stokes equations. The proposed scheme deals with the spatial discretization by Galerkin finite element method (FEM), treats the temporal discretization by CNLF method for the linear term and the semi-implicit method for nonlinear term. The almost unconditional stability, i.e., the time step is no more than a constant, is proven. By a new negative norm technique, the L 2 -optimal error estimates with respect to temporal and spacial orientation for the velocity are derived. At last, some numerical results are provided to justify our theoretical analysis.

Authors:Falletta Monegato; Scuderi Abstract: Publication date: February 2018 Source:Applied Numerical Mathematics, Volume 124 Author(s): S. Falletta, G. Monegato, L. Scuderi In this paper, we consider 3D wave propagation problems in unbounded domains, such as those of acoustic waves in non viscous fluids, or of seismic waves in (infinite) homogeneous isotropic materials, where the propagation velocity c is much higher than 1. For example, in the case of air and water c ≈ 343 m / s and c ≈ 1500 m / s respectively, while for seismic P-waves in linear solids we may have c ≈ 6000 m / s or higher. These waves can be generated by sources, possible away from the obstacles. We further assume that the dimensions of the obstacles are much smaller than that of the wave velocity, and that the problem transients are not excessively short. For their solution we consider two different approaches. The first directly uses a known space–time boundary integral equation to determine the problem solution. In the second one, after having defined an artificial boundary delimiting the region of computational interest, the above mentioned integral equation is interpreted as a non reflecting boundary condition to be coupled with a classical finite element method. For such problems, we show that in some cases the computational cost and storage, required by the above numerical approaches, can be significantly reduced by taking into account a property that till now has not been considered. To show the effectiveness of this reduction, the proposed approach is applied to several problems, including multiple scattering.

Authors:Xue Luo Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Xue Luo In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments, under some conditions. The numerical experiments of the inviscid Burger's equation support our result, and it verifies the reasonableness of the conditions.

Authors:Carlos Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Carlos Pérez-Arancibia This paper presents expressions for the classical combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems on the plane, in terms of smooth (continuously differentiable) integrands. These expressions are obtained by means of a singularity subtraction technique based on pointwise plane-wave expansions of the unknown density function. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. Moreover, the expressions for the combined field integral operators and layer potentials presented in this contribution can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners—one of which is based on direct application of the trapezoidal rule—demonstrates the effectiveness of the proposed higher-order singularity subtraction approach.

Authors:Vít Dolejší; Georg May; Ajay Rangarajan Abstract: Publication date: Available online 2 October 2017 Source:Applied Numerical Mathematics Author(s): Vít Dolejší, Georg May, Ajay Rangarajan We present a continuous-mesh model for anisotropic hp-adaptation in the context of numerical methods using discontinuous piecewise polynomial approximation spaces. The present work is an extension of a previously proposed mesh-only (h-)adaptation method which uses both a continuous mesh, and a corresponding high-order continuous interpolation operator. In this previous formulation local anisotropy and global mesh density distribution may be determined by analytical optimization techniques, operating on the continuous mesh model. The addition of varying polynomial degree necessitates a departure from purely analytic optimization. However, we show in this article that a global optimization problem may still be formulated and solved by analytic optimization, adding only the necessity to solve numerically a single nonlinear algebraic equation per adaptation step to satisfy a constraint on the total number of degrees of freedom. The result is a tailorsuited continuous mesh with respect to a model for the global interpolation error measured in the L q -norm. From the continuous mesh a discrete triangular mesh may be generated using any metric-based mesh generator.

Authors:Simon Becher Abstract: Publication date: Available online 21 September 2017 Source:Applied Numerical Mathematics Author(s): Simon Becher We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted piecewise equidistant meshes proposed by Sun and Stynes. We also study the streamline-diffusion finite element method (SDFEM) for such problems. For these methods error estimates uniform with respect to ε are proven in the energy norm and in the stronger SDFEM-norm, respectively. Numerical experiments confirm the theoretical findings.

Authors:Michal Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): Michal Beneš We propose and rigorously analyze the subcycling method based on primal domain decomposition techniques for first-order transient partial differential equations. In time dependent problems, it can be computationally advantageous to use different time steps in different regions. Smaller time steps are used in regions of significant changes in the solution and larger time steps are prescribed in regions with nearly stationary response. Subcycling can efficiently reduce the total computational cost. Crucial to our approach is a nonstandard heterogeneous temporal discretization. We begin with the discretization in time by the asynchronous Rothe method, which, in essence, involves a backward finite difference scheme assuming different time steps (fine and large time steps) in different parts of the computational domain. The emphasis of the paper is on qualitative properties of the new numerical scheme, such as a-priori estimates, existence of the time-discrete solutions and the strong convergence and stability analysis. Several numerical experiments were conducted to examine the consistency of the proposed method.

Authors:Mokhtary Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): P. Mokhtary The main concern of this paper is to develop and analyze an operational Tau method for obtaining the numerical solution of fractional weakly singular integro-differential equations when the Jacobi polynomials are used as natural basis functions. This strategy is an application of the matrix–vector–product approach in Tau formulation of the problem. We first study the regularity of the exact solution and show that some derivatives of the exact solution have a singularity at origin dependence on both order of fractional derivative and weakly singular kernel function which makes poor convergence results for the Tau discretization of the problem. In order to recover high-order of convergence, we propose a new variable transformation to regularize the given functions and then to approximate the solution via a satisfactory order of convergence using an operational Tau method. Convergence analysis of this novel method is presented and the expected spectral rate of convergence for the proposed method is established. Numerical results are given which confirm both the theoretical predictions obtained and efficiency of the proposed method.

Abstract: Publication date: November 2017 Source:Applied Numerical Mathematics, Volume 121 Author(s): K. Šišková, M. Slodička In the present paper, we deal with an inverse source problem for a time-fractional wave equation in a bounded domain in R d . The time-dependent source is determined from an additional measurement in the form of integral over the space subdomain. The existence, uniqueness and regularity of a weak solution are obtained. A numerical algorithm based on Rothe's method is proposed, a priori estimates are proved and convergence of iterates towards the solution is established.