Authors:Jing An; Jie Shen; Zhimin Zhang Pages: 1 - 15 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Jing An, Jie Shen, Zhimin Zhang In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form − div ( A ∇ u ) + V u + f ( u 2 ) u = λ u , ‖ u ‖ L 2 = 1 . We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.
Authors:E. Aulisa; G. Bornia; S. Calandrini; G. Capodaglio Pages: 16 - 38 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): E. Aulisa, G. Bornia, S. Calandrini, G. Capodaglio In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs under no regularity assumptions on the solution of the problem. The proposed analysis provides three main contributions to the existing theory. The first novel contribution of this study is a convergence bound that depends on the number of multigrid smoothing iterations. This result is obtained under no regularity assumptions on the solution of the problem. A similar result has been shown in the literature for the cases of full regularity and partial regularity assumptions. Second, our theory applies to local refinement applications with arbitrary level hanging nodes. More specifically, for the smoothing algorithm we provide subspace decompositions that are suitable for applications where the multigrid spaces are defined on finite element grids with arbitrary level hanging nodes. Third, global smoothing is employed on the entire multigrid space with hanging nodes. When hanging nodes are present, existing multigrid strategies advise to carry out the smoothing procedure only on a subspace of the multigrid space that does not contain hanging nodes. However, with such an approach, if the number of smoothing iterations is increased, convergence can improve only up to a saturation value. Global smoothing guarantees an arbitrary improvement in the convergence when the number of smoothing iterations is increased. Numerical results are also included to support our theoretical findings.
Authors:Xin Li; Luming Zhang Pages: 39 - 53 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Xin Li, Luming Zhang In this article, a trigonometric integrator sine pseudo-spectral (TISP) method is presented for the extended Fisher–Kolmogorov equation. This method depends on a Gautschi-type integrator in phase space to the temporal approximation and the sine pseudo-spectral method to the spatial discretization. Rigorous error estimates are carried out in the energy space by utilizing the mathematical induction. The error bound shows the new scheme which established by the TISP method has second-order accurate in time and spectral-order accurate in space. Moreover, the new scheme is generalized to higher dimensions. The compact finite difference (CFD) scheme in one and two dimensions which supported by the method of order reduction are constructed as a benchmark for comparisons. Comparison results between two schemes are given to confirm the theoretical studies and demonstrate the efficiency and accuracy of TISP method in both one and multi-dimensional problems.
Authors:Denys Dutykh; Jean-Guy Caputo Pages: 54 - 71 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Denys Dutykh, Jean-Guy Caputo We consider a scalar Hamiltonian nonlinear wave equation formulated on networks; this is a non standard problem because these domains are not locally homeomorphic to any subset of the Euclidean space. More precisely, we assume each edge to be a 1D uniform line with end points identified with graph vertices. The interface conditions at these vertices are introduced and justified using conservation laws and an homothetic argument. We present a detailed methodology based on a symplectic finite difference scheme together with a special treatment at the junctions to solve the problem and apply it to the sine-Gordon equation. Numerical results on a simple graph containing four loops show the performance of the scheme for kinks and breathers initial conditions.
Authors:Z. Soori; A. Aminataei Pages: 72 - 94 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Z. Soori, A. Aminataei In this paper, we propose a high-order scheme for the numerical solution of multi-term time fractional diffusion-wave (FDW) equation in one and two-dimensional on non-uniform grids. Based on the sixth-order non-uniform combined compact difference (NCCD) scheme in the space directions on non-uniform grids, an alternating direction implicit (ADI) method is proposed to split the equation into two separate one dimensional equations. The multi-term time fractional derivation is described in the Caputo's sense with scheme of order O ( τ 3 − α ) , 1 < α < 2 . A numerical analysis of Fourier analysis completed by stability calculations in terms of semi-discrete eigenvalue problems are proposed. The advantage of the non-uniform combined compact difference (NCCD) scheme is that it can decrease the CPU time in comparison with the uniform combined compact difference (CCD) scheme. The sixth-order accuracy in the space directions on non-uniform grids has not been achieved in previous studies.
Authors:Dongyang Shi; Huaijun Yang Pages: 109 - 122 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Dongyang Shi, Huaijun Yang A new low order nonconforming mixed finite element method (MFEM) is proposed and analyzed for time-fractional diffusion equation with element pair (CNR Q 1 + Q 0 × Q 0 ). A new error estimate for the consistency error of nonconforming element CNR Q 1 is proved, which leads to the superclose and superconvergence results of the original variable in broken H 1 norm, and of the flux in L 2 norm for a fully-discrete scheme with the Caputo derivative approximated by the classical L1 method. The results obtained herein improve the corresponding conclusions in the previous literature. Finally, some numerical results are provided to confirm the theoretical analysis.
Authors:Xiaoli Li; Hongxing Rui Pages: 123 - 139 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Xiaoli Li, Hongxing Rui In this article, a block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation with Neumann boundary condition is introduced and analyzed. The unconditional stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O ( Δ t 1 + σ / 2 + h 2 + k 2 + σ 2 ) both for pressure and velocity are established on non-uniform rectangular grids, where Δ t , h , k and σ are the step sizes in time, space in x- and y-direction, and distributed order. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
Authors:Pouria Assari; Mehdi Dehghan Pages: 140 - 157 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Pouria Assari, Mehdi Dehghan In this investigation, a computational scheme is given to solve nonlinear one- and two-dimensional Volterra integral equations of the second kind. We utilize the radial basis functions (RBFs) constructed on scattered points by combining the discrete collocation method to estimate the solution of Volterra integral equations. All integrals appeared in the scheme are approximately computed by the composite Gauss–Legendre integration formula. The implication of previous methods for solving these types of integral equations encounters difficulties by increasing the dimensional of problems and sometimes requires a mesh generation over the solution region. While the new technique presented in the current paper does not increase the difficulties for higher dimensional integral equations due to the easy adaption of RBF and also needs no cell structures on the domains. Moreover, we obtain the error bound and the convergence rate of the proposed approach. Illustrative examples clearly show the reliability and efficiency of the method and confirm the theoretical error estimates.
Authors:Ahmet Guzel; Catalin Trenchea Pages: 158 - 173 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Ahmet Guzel, Catalin Trenchea The explicit weakly-stable second-order accurate leapfrog scheme is widely used in the numerical models of weather and climate, in conjunction with the Robert–Asselin (RA) and Robert–Asselin–Williams (RAW) time filters. The RA and RAW filters successfully suppress the spurious computational mode associated with the leapfrog method, but also weakly damp the physical mode and degrade the numerical accuracy to first-order. The recent higher-order Robert–Asselin (hoRA) time filter reduces the undesired numerical damping of the RA and RAW filters and increases the accuracy to second up-to third-order. We prove that the combination of leapfrog-hoRA and Williams' step increases the stability by 25%, improves the accuracy of the amplitude of the physical mode up-to two significant digits, effectively suppresses the computational modes, and further diminishes the numerical damping of the hoRA filter.
Authors:S. Nemati; P. Lima; S. Sedaghat Pages: 174 - 189 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): S. Nemati, P. Lima, S. Sedaghat In this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of convergence order O ( h 3 ) .
Authors:Mehdi Dehghan; Mostafa Abbaszadeh Pages: 190 - 206 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Mehdi Dehghan, Mostafa Abbaszadeh The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6]. The temporal direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains.
Authors:M. Braś; A. Cardone; Z. Jackiewicz; P. Pierzchała Pages: 207 - 231 Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): M. Braś, A. Cardone, Z. Jackiewicz, P. Pierzchała We consider the class of implicit–explicit general linear methods (IMEX). Such schemes are designed for ordinary differential equation systems with right hand side function splitted into stiff and non-stiff parts. We investigate error propagation of IMEX methods up to the terms of order p + 2 . In addition, we construct IMEX schemes of order p and stage order q = p , p ≤ 4 and we verify the performance of methods in several numerical experiments.
Authors:Rui Zhan; Jingjun Zhao Pages: 1 - 22 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Rui Zhan, Jingjun Zhao In this paper, the convergence analysis of operator splitting methods for the Camassa–Holm equation is provided. The analysis is built upon the regularity of the Camassa–Holm equation and the divided equations. It is proved that the solution of the Camassa–Holm equation satisfies the locally Lipschitz condition in H 1 and H 2 norm, which ensures the regularity of the numerical solution. Through the calculus of Lie derivatives, we show that the Lie–Trotter and Strang splitting converge with the expected rate under suitable assumptions. Numerical experiments are presented to illustrate the theoretical result.
Authors:Jana Burkotová; Irena Rachůnková; Svatoslav Staněk; Ewa B. Weinmüller; Stefan Wurm Pages: 23 - 50 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Jana Burkotová, Irena Rachůnková, Svatoslav Staněk, Ewa B. Weinmüller, Stefan Wurm We study boundary value problems for systems of nonlinear ordinary differential equations with a time singularity, x ′ ( t ) = M ( t ) t x ( t ) + f ( t , x ( t ) ) t , t ∈ ( 0 , 1 ] , b ( x ( 0 ) , x ( 1 ) ) = 0 , where M : [ 0 , 1 ] → R n × n and f : [ 0 , 1 ] × R n → R n are continuous matrix-valued and vector-valued functions, respectively. Moreover, b : R n × R n → R n is a continuous nonlinear mapping which is specified according to a spectrum of the matrix M ( 0 ) . For the case that M ( 0 ) has eigenvalues with nonzero real parts, we prove new results about existence of at least one continuous solution on the closed interval [ 0 , 1 ] including the singular point, t = 0 . We also formulate sufficient conditions for uniqueness. The theory is illustrated by a numerical simulation based on the collocation method.
Authors:Roland Pulch; Diana Estévez Schwarz; René Lamour Pages: 51 - 69 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Roland Pulch, Diana Estévez Schwarz, René Lamour In radio frequency applications, electric circuits generate signals, which are amplitude modulated and/or frequency modulated. A mathematical modelling typically yields systems of differential algebraic equations (DAEs). A multivariate signal model transforms the DAEs into multirate partial differential algebraic equations (MPDAEs). In the case of frequency modulation, an additional condition is required to identify an appropriate solution. We consider a necessary condition for an optimal solution and a phase condition. A method of lines, which discretises the MPDAEs as well as the additional condition, generates a larger system of DAEs. We analyse the differentiation index of this approximate DAE system, where the original DAEs are assumed to be semi-explicit systems. The index depends on the inclusion of either differential variables or algebraic variables in the additional condition. We present results of numerical simulations for an illustrative example, where the index is also verified by a numerical method.
Authors:Behnam Soleimani; Rüdiger Weiner Pages: 70 - 85 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Behnam Soleimani, Rüdiger Weiner In this paper we will focus on numerical methods for differential equations with both stiff and nonstiff parts. This kind of problems can be treated efficiently by implicit-explicit (IMEX) methods and here we investigate a class of s-stage IMEX peer methods of order p = s for variable and p = s + 1 for constant step sizes. They are combinations of s-stage superconvergent implicit and explicit peer methods. We construct methods of order p = s + 1 for s = 3 , 4 , 5 where we compute the free parameters numerically to give good stability with respect to a general linear test problem frequently used in the literature. Numerical comparisons with two-step IMEX Runge–Kutta methods confirm the high potential of the new constructed superconvergent IMEX peer methods.
Authors:Mikhail Bulatov; Liubov Solovarova Pages: 86 - 94 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Mikhail Bulatov, Liubov Solovarova In this article a class of linear differential–algebraic equations with an initial condition is identified. This class has a unique continuously differentiable solution that depends on the first derivatives of the right-hand part. Assuming that the right-hand part is given with the known level of the error, it is shown that a difference scheme of the first order generates a regularization algorithm. The integration step that depends on the perturbation of the right-hand part is the regularization parameter. The survey of regularization methods for differential–algebraic equations and related problems is given.
Authors:Xiu Yang; Xiaoyun Jiang; Hui Zhang Pages: 95 - 111 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Xiu Yang, Xiaoyun Jiang, Hui Zhang This paper is devoted to the numerical solution of the time fractional cable equation and its inverse problem. The time–space spectral Legendre tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem. Furthermore, we prove that the approximated solution of this method converges to the exact solution. In addition, the inverse problem is formulated by using the Tikhonov regularization, the stability and convergence for the inverse problem are provided, then we analyse the sensitivity coefficients and apply the nonlinear conjugate gradient method to solve the regularized problem. Finally, some numerical results are carried out to support the theoretical claims.
Authors:Haifeng Ji; Jinru Chen; Zhilin Li Pages: 112 - 130 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Haifeng Ji, Jinru Chen, Zhilin Li A high-order finite element method based on unfitted meshes for solving a class of elliptic interface problems whose solution and its normal derivative have finite jumps across an interface is proposed in this paper. The idea of the method is based on the source removal technique first introduced in the immersed interface method (IIM). The strategy is to use the level set representation of the interface and extend the jump conditions that are defined along the interface to a neighborhood of the interface. In our numerical method, the jump conditions only need to be extended to the Lagrange points of elements intersecting with the interface. Optimal error estimates of the method in the broken H 1 and L 2 norms are rigorously proven. Numerical examples presented in this paper also confirm our theoretical analysis.
Authors:Scott A. Sarra; Yikun Bai Pages: 131 - 142 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Scott A. Sarra, Yikun Bai Radial Basis Function (RBF) methods have become important tools for scattered data interpolation and for solving partial differential equations (PDEs) in complexly shaped domains. When the underlying function is sufficiently smooth, RBF methods can produce exceptional accuracy. However, like other high order numerical methods, if the underlying function has steep gradients or discontinuities the RBF method may/will produce solutions with non-physical oscillations. In this work, a rational RBF method is used to approximate derivatives of functions with steep gradients and discontinuities and to solve PDEs with such solutions. The method is non-linear and is more computationally expensive than the standard RBF method. A modified partition of unity method is discussed as an way to implement the rational RBF method in higher dimensions.
Authors:Karel J. in 't Hout; Jari Toivanen Pages: 143 - 156 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Karel J. in 't Hout, Jari Toivanen This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations.
Authors:Qian Guo; Wei Liu; Xuerong Mao Pages: 157 - 170 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Qian Guo, Wei Liu, Xuerong Mao The partially truncated Euler–Maruyama (EM) method was recently proposed in our earlier paper [3] for highly nonlinear stochastic differential equations (SDEs), where the finite-time strong L r -convergence theory was established. In this note, we will point out that one condition imposed there is restrictive in the sense that this condition might force the stepsize to be so small that the partially truncated EM method would be inapplicable. In this note, we will remove this restrictive condition but still be able to establish the finite-time strong L r -convergence rate. The advantages of our new results will be highlighted by the comparisons with our earlier results in [3].
Authors:Tingting Zhang; Hui Liang Pages: 171 - 183 Abstract: Publication date: August 2018 Source:Applied Numerical Mathematics, Volume 130 Author(s): Tingting Zhang, Hui Liang The multistep collocation method is applied to Volterra integral equations of the first kind. The existence and uniqueness of the multistep collocation solution are proved. Then the convergence condition of the multistep collocation method is analyzed and the corresponding convergence order is described. In particular, for c m = 1 , the convergence conditions, which can be easily implemented, are given for two-step and three-step collocation methods. Numerical experiments illustrate the theoretical analysis.
Authors:Elyas Shivanian; Ahmad Jafarabadi Pages: 1 - 25 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Elyas Shivanian, Ahmad Jafarabadi In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse source problem of time-fractional diffusion equation in two dimensions. The missing solely time-dependent source is recovered from an additional integral measurement. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of SMRPI. Firstly, we use a difference scheme for the fractional derivative to discretize the governing equation, and we obtain a finite difference scheme with respect to time. Then we use the SMRPI approach to approximate the spatial derivatives. Also, it is proved that the scheme is unconditionally stable with respect to time in space H 1 . Consequently, when the input data is contaminated with noise, we use the Tikhonov regularization method in order to obtain a stable solution. Numerical results show that the solution is accurate for exact data and stable for noisy data.
Authors:Jingliang Li; Heping Ma; Yonghui Qin; Shuaiyin Zhang Pages: 26 - 38 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Jingliang Li, Heping Ma, Yonghui Qin, Shuaiyin Zhang In this paper, we study a spectral method for the triangular prism. We construct an approximation space in the “pole” condition in which the integral singularity is removed in a simple and effective way. We build a quasi-interpolation operator in the approximation space, and analyze its L 2 -error. Based on the quasi-interpolation, a triangular prism spectral method for the elliptic modal problem is studied. Furthermore, we extend this triangular prism spectral method to a triangular prism spectral element method. For the elliptic modal problem, we present the spectral element scheme and analyze the convergence. At last, we do some experiments to test the effectiveness of the method.
Authors:Saulo Pomponet Oliveira; Stela Angelozi Leite Pages: 39 - 57 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Saulo Pomponet Oliveira, Stela Angelozi Leite We present the error analysis of a high-order method for the two-dimensional acoustic wave equation in the particular case of constant compressibility and variable density. The domain discretization is based on the spectral element method with Gauss–Lobatto–Legendre (GLL) collocation points, whereas the time discretization is based on the explicit leapfrog scheme. As usual, GLL points are also employed in the numerical quadrature, so that the mass matrix is diagonal and the resulting algebraic scheme is explicit in time. The analysis provides an a priori estimate which depends on the time step, the element length, and the polynomial degree, generalizing several known results for the wave equation in homogeneous media. Numerical examples illustrate the validity of the estimate under certain regularity assumptions and provide expected error estimates when the medium is discontinuous.
Authors:Maohua Ran; Chengjian Zhang Pages: 58 - 70 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Maohua Ran, Chengjian Zhang In this paper, a class of new compact difference schemes is presented for solving the fourth-order time fractional sub-diffusion equation of the distributed order. By using an effective numerical quadrature rule based on boundary value method to discretize the integral term in the distributed-order derivative, the original distributed order differential equation is approximated by a multi-term time fractional sub-diffusion equation, which is then solved by a compact difference scheme. It is shown that the suggested compact difference scheme is stable and convergent in L ∞ norm with the convergence order O ( τ 2 + h 4 + ( Δ γ ) p ) when a boundary value method of order p is used, where τ , h and Δγ are the step sizes in time, space and distributed-order variables, respectively. Numerical results are reported to verify the high order accuracy and efficiency of the suggested scheme. Moreover, in the example, comparisons between some existing methods and the suggested scheme is also provided, showing that our method doesn't compromise in computational time.
Authors:Yunyun Ma; Fuming Ma Pages: 71 - 82 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Yunyun Ma, Fuming Ma We develop a projection method with regularization for reconstructing the radiation electromagnetic field in the exterior of a bounded domain from the knowledge of Cauchy data. The method is divided into two parts. We first solve the complete tangential component of the electrical field on the boundary of that domain from Cauchy data. The radiation electromagnetic field is then recovered from the complete tangential component of the electrical field. For the first part, we transform the Cauchy problem into a compact operator equation by means of the electric-to-magnetic Calderón operator and propose a projection method with regularization to solve that compact operator equation. Meanwhile, we analyze the asymptotic behavior of the singular values of the corresponding compact operator. For the second part, we expend the radiation electromagnetic field to the vector spherical harmonics. Numerical examples are finally presented to demonstrate the computational efficiency of the proposed method.
Authors:Tong Zhang; Yanxia Qian; JinYun Yuan Pages: 83 - 103 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Tong Zhang, Yanxia Qian, JinYun Yuan In this paper, we consider the stability and convergence results of numerical solutions in fully discrete fractional-step formulation for the Oldroyd model. The proposed numerical method is constructed by the decompositions of the viscosity in time part and the finite element method in space part. With some mild regularity assumptions on the exact solution, the unconditional stability results of the approximate solutions are established. Then, the first order spatial convergence for the “end-of-step” velocity is shown. Based on the above results, the optimal order approximations for the velocity and pressure in L 2 -norm are obtained for the mesh size. Finally, two numerical examples are given to illustrate the established theoretical analysis and test the performances of the developed numerical method and show the influences of the memory term for the considered problem.
Authors:Roman Chapko; B. Tomas Johansson Pages: 104 - 119 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Roman Chapko, B. Tomas Johansson We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogeneous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach.
Authors:Jialin Hong; Chuying Huang; Xu Wang Pages: 120 - 136 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Jialin Hong, Chuying Huang, Xu Wang We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter H ∈ ( 1 / 4 , 1 / 2 ] . We prove that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge–Kutta methods, which are implicit methods in general. If the vector fields satisfy some smoothness and boundedness conditions, we obtain the pathwise convergence rates of Runge–Kutta methods. When vector fields are linear, we get the solvability of the midpoint scheme for skew symmetric cases, and obtain its pathwise convergence rate. Numerical experiments verify our theoretical analysis.
Authors:Pengde Wang; Chengming Huang Pages: 137 - 158 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Pengde Wang, Chengming Huang This paper considers the long-time integration of the nonlinear fractional Schrödinger equation involving the fractional Laplacian from the point of view of symplectic geometry. By virtue of a variational principle with the fractional Laplacian, the equation is first reformulated as a Hamiltonian system with a symplectic structure. Then, by introducing a pair of intermediate variables with a fractional operator, the equation is reformulated in another form for which more conservation laws are found. When reducing to the case of integer order, they correspond to multi-symplectic conservation law and local energy conservation law for the classic Schrödinger equation. After that, structure-preserving algorithms with the Fourier pseudospectral approximation to the spatial fractional operator are constructed. It is proved that the semi-discrete and fully discrete systems satisfy the corresponding symplectic or other conservation laws in the discrete sense. Numerical tests are performed to validate the efficiency of the methods by showing their remarkable conservation properties in the long-time simulation.
Authors:E.N.G. Grylonakis; C.K. Filelis-Papadopoulos; G.A. Gravvanis Pages: 159 - 180 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): E.N.G. Grylonakis, C.K. Filelis-Papadopoulos, G.A. Gravvanis A class of numerical techniques is introduced for computing the solution and its normal derivative for two-dimensional, linear, elliptic partial differential equations (PDEs) in the interior of convex polygonal domains. The cornerstone of this approach is the Fokas approximate global relation, i.e. a discretized equation on the complex k-plane that couples the integral transforms of the given and the unknown boundary data, derived by analyzing the corresponding boundary value problem (BVP) with the Fokas unified transform method. The proposed techniques rely on the partitioning of a given polygonal computational domain into a number of concentric, polygonal subdomains. On each subdomain a global relation holds, and furthermore, the Dirichlet and Neumann values between consecutive polygons are related via prescribed algebraic relations. By assembling all the available equations into a linear system, it is possible to compute the solution as well as its normal derivative over the whole domain by solving a number of smaller, one-dimensional BVPs simultaneously. The advantage of the proposed methodology is the simultaneous approximation of both the solution and the normal derivative over the entire of a general polygonal domain with satisfactory accuracy, by solving one sparse linear system. In addition, our suggested approach provides a framework for designing different numerical schemes with prescribed order of accuracy, whereas it is possible to implement adaptive techniques in terms of the mesh size and the order of the expansion polynomials. Here, we present the details of the proposed methodology and provide numerical results for solving linear elliptic PDEs on convex polygons.
Authors:Sanjar M. Abrarov; Brendan M. Quine; Rajinder K. Jagpal Pages: 181 - 191 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Sanjar M. Abrarov, Brendan M. Quine, Rajinder K. Jagpal Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function w ( z ) = e − z 2 ( 1 + 2 i π ∫ 0 z e t 2 d t ) , where z = x + i y . As a further development, in this work we show how this sampling-based rational approximation can be transformed into alternative form for efficient computation of the complex error function w ( z ) at smaller values of the imaginary argument y = Im [ z ] . Such an approach enables us to avoid poles in implementation and to cover the entire complex plain with high accuracy in a rapid algorithm. An optimized Matlab code utilizing only three rapid approximations is presented.
Authors:Hendrik Ranocha; Jan Glaubitz; Philipp Öffner; Thomas Sonar Pages: 1 - 23 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Hendrik Ranocha, Jan Glaubitz, Philipp Öffner, Thomas Sonar The flux reconstruction is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms, artificial dissipation/spectral viscosity operators and connections to modal filtering are investigated. Firstly, the discrete viscosity operators are studied for general SBP bases, stressing the importance of the correct implementation in order to get both conservative and stable approximations. Starting from L 2 stability for scalar conservation laws, the results are extended to entropy stability for hyperbolic systems and supported by numerical experiments. Furthermore, the relation to modal filtering is recalled and several possibilities to apply filters are investigated, both analytically and numerically. This analysis serves to single out a unique way to apply modal filters in numerical methods.
Authors:Mahboub Baccouch Pages: 43 - 64 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Mahboub Baccouch The local discontinuous Galerkin (LDG) method has been successfully applied to deterministic boundary-value problems (BVPs) arising from a wide range of applications. In this paper, we propose a stochastic analogue of the LDG method for stochastic two-point BVPs. We first approximate the white noise process by a piecewise constant random process to obtain an approximate BVP. We show that the solution of the new BVP converges to the solution of the original problem. The new problem is then discretized using the LDG method for deterministic problems. We prove that the solution to the new approximate BVP has better regularity which facilitates the convergence proof for the proposed LDG method. More precisely, we prove L 2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be two in the mean-square sense, when piecewise polynomials of degree at most p are used. Finally, several numerical examples are provided to illustrate the theoretical results.
Authors:Lunji Song; Shan Zhao; Kaifang Liu Pages: 65 - 80 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Lunji Song, Shan Zhao, Kaifang Liu A new relaxed weak Galerkin (WG) stabilizer has been introduced for second order elliptic interface problems with low regularity solutions. The stabilizer is generalized from the weak Galerkin method of Wang and Ye by using a new relaxation index to mesh size and the index β can be tuned according to the regularity of solution. The relaxed stabilization gives rise to a considerable flexibility in treating weak continuity along interior element edges and interface edges. For solutions in Sobolev space W l + 1 , p , with l ≥ 0 and p ∈ ( 1 , 2 ] rather than the usual case p = 2 , we derive convergence orders of the new WG method in the energy and L p norms under some regularity assumptions of the solution and an optimal selection of β = 1 + 4 p − p can be given in the energy norm. It is recovered for p = 2 that with the choice of β = 1 , error estimates in the energy and L 2 norms are optimal for the source term in the sobolev space L 2 . The stabilized WG method can be easily implemented without requiring any sufficiently large penalty factor. In addition, numerical results demonstrate the effectiveness and optimal convergence of the proposed WG method with an over-relaxed factor β.
Authors:André Pierro de Camargo Pages: 81 - 83 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): André Pierro de Camargo We show, by elementary calculations, that the condition number (with respect to the supremum norm) of every n × n real Vandermonde matrix is at least 2 n − 2 .
Authors:Thi Thanh Mai Ta; Van Chien Le; Ha Thanh Pham Pages: 160 - 179 Abstract: Publication date: July 2018 Source:Applied Numerical Mathematics, Volume 129 Author(s): Thi Thanh Mai Ta, Van Chien Le, Ha Thanh Pham
Abstract: Publication date: Available online 15 June 2018 Source:Applied Numerical Mathematics Author(s): Lukas Einkemmer The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used. In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion–advection equation, Burgers' equation with a reaction term, porous media equation, viscous Burgers' equation, Allen–Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.
Abstract: Publication date: Available online 15 June 2018 Source:Applied Numerical Mathematics Author(s): R. Čiegis, O. Suboč In this paper we consider high-order compact finite difference schemes constructed on 1D non-uniform grids. We apply them to parabolic and Schrödinger equations. Stability of these schemes is investigated by using the spectral method. Computer experiments are applied in order to find critical grids for which the stability condition is violated. Such grids are obtained for the Schrödinger problem, but not for the parabolic problems. Numerical examples supporting our theoretical analysis are provided and discussed.
Abstract: Publication date: Available online 14 June 2018 Source:Applied Numerical Mathematics Author(s): Shi-Liang Wu, Peng Guo In this paper, a class of modulus-based matrix splitting iteration methods for the quasi-complementarity problems is presented. The convergence analysis of the proposed methods is discussed. Numerical experiments show that the proposed methods are efficient.
Authors:Tianliang Hou; Luoping Chen; Yin Yang Abstract: Publication date: Available online 7 June 2018 Source:Applied Numerical Mathematics Author(s): Tianliang Hou, Luoping Chen, Yin Yang In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart-Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz-Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h = H 2 , the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.
Authors:Yongbing Luo; Yanbing Yang Salik Ahmed Tao Mingyou Zhang Ligang Abstract: Publication date: Available online 6 June 2018 Source:Applied Numerical Mathematics Author(s): Yongbing Luo, Yanbing Yang, Md Salik Ahmed, Tao Yu, Mingyou Zhang, Ligang Wang, Huichao Xu This paper investigates the local existence, global existence and finite time blow up of the solution to the Cauchy problem for a class of nonlinear Klein-Gordon equation with general power-type nonlinearities. We give some sufficient conditions on the initial data such that the solution exists globally or blows up in finite time with low initial energy and critical energy. Further a finite time blow up result of the solution with high initial energy is proved.
Authors:Toshihiro Yamada Abstract: Publication date: September 2018 Source:Applied Numerical Mathematics, Volume 131 Author(s): Toshihiro Yamada This paper shows a discretization method of solution to stochastic differential equations as an extension of the Milstein scheme. With a simple method, we reconstruct weak Milstein scheme through second order polynomials of Brownian motions without assuming the Lie bracket commutativity condition on vector fields imposed in the classical Milstein scheme and show a sharp error bound for it. Numerical example illustrates the validity of the scheme.
Authors:Andrzej Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Andrzej Kałuża, Paweł Przybyłowicz We provide a construction of an implementable method based on path-independent adaptive step-size control for global approximation of jump-diffusion SDEs. The sampling points are chosen in nonadaptive way with respect to trajectories of the driving Poisson and Wiener processes. However, they are adapted to the diffusion and jump coefficients of the underlying stochastic differential equation and to the values of intensity function of the driving Poisson process. The method is asymptotically optimal in the class of methods that use (possibly) non-equidistant discretization of the interval [ 0 , T ] and is more efficient than any method based on the uniform mesh.
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