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: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:Jijun Liu; Bingxian Wang Pages: 84 - 97 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Jijun Liu, Bingxian Wang The backward heat conduction problems aim to determine the temperature distribution in the past time from the present measurement data. For this linear ill-posed problem, we propose a homotopy-based iterative regularizing scheme for noisy input data. The advantages of the proposed scheme are, under general assumptions on the exact initial distribution, we can always ensure the convergence of the homotopy sequence with exact final data as initial guess. For noisy input data, we also establish the error analysis for the regularizing solution with noisy measurement data as our initial guess. Our algorithm is easily implementable with very low computational costs in the sense that we only need to do one iteration from initial guess using the final noisy data directly, while the error is still comparable to other regularizing methods. Numerical implementations are presented.

Authors:Kareem T. Elgindy; Hareth M. Refat Pages: 98 - 124 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Kareem T. Elgindy, Hareth M. Refat We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudo-spectral method (SGIPSM) to solve numerically Lane–Emden equations with mixed Neumann and Robin boundary conditions. The framework of the proposed method includes: (i) recasting the problem into its integral formulation, (ii) collocating the latter at the shifted flipped-Gegenbauer–Gauss–Radau (SFGGR) points, and (iii) replacing the integrals with accurate and well-conditioned numerical quadratures constructed via SFGGR-based shifted Gegenbauer integration matrices. The integral formulation is eventually discretized into linear/nonlinear system of equations that can be solved easily using standard direct system solvers. The implementation of the proposed method is further illustrated through four efficient computational algorithms. The theoretical study is enriched with rigorous error, convergence, and stability analyses of the SGIPSM. The paper highlights some interesting new findings pertaining to “the apt choice of Gegenbauer collocation set of points” that could largely influence the proper use of Gegenbauer polynomials as basis polynomials for polynomial interpolation and collocation. Five numerical test examples are presented to verify the effectiveness, accuracy, exponential convergence, and numerical stability of the proposed method. The numerical simulations are associated with extensive numerical comparisons with other rival methods in the literature to demonstrate further the power of the proposed method. The SGIPSM is broadly applicable and represents a strong addition to common numerical methods for solving linear/nonlinear differential equations when high-order approximations are required using a relatively small number of collocation points.

Authors:Zhihao Ge; Mengxia Ma Pages: 125 - 138 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Zhihao Ge, Mengxia Ma In the paper, we propose a multirate iterative scheme based on multiphysics discontinuous Galerkin method for a poroelasticity model. We solve a generalized Stokes problem in the coarse time size and solve the diffusion problem in the finer time size, and use the multiphysics discontinuous Galerkin method for the discretization of the space variables. And we prove that the multirate iterative scheme is stable and the numerical solution satisfies some energy conservation laws. Also, we prove that the convergence order is optimal in the energy norm. Finally, we give the numerical tests to verify that the multirate iterative scheme not only greatly reduces the computational cost, but also has no “locking phenomenon”.

Authors:Lizhen Chen; Jia Zhao; Xiaofeng Yang Pages: 139 - 156 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Lizhen Chen, Jia Zhao, Xiaofeng Yang In this paper, we propose full discrete linear schemes for the molecular beam epitaxy (MBE) model with slope selection, which are shown to be unconditionally energy stable and unique solvable. In details, using the invariant energy quadratization (IEQ) approach, along with a regularized technique, the MBE model is first discretized in time using either Crank–Nicolson or Adam–Bashforth strategies. The semi-discrete schemes are shown to be energy stable and unique solvable. Then we further use Fourier-spectral methods to discretize the space, ending with full discrete schemes that are energy-stable and unique solvable. In particular, the full discrete schemes are linear such that only a linear algebra problem need to be solved at each time step. Through numerical tests, we have shown a proper choice of the regularization parameter provides better stability and accuracy, such that larger time step is feasible. Afterward, we present several numerical simulations to demonstrate the accuracy and efficiency of our newly proposed schemes. The linearizing and regularizing strategy used in this paper could be readily applied to solve a class of phase field models that are derived from energy variation.

Authors:Farhad Fakhar-Izadi; Mehdi Dehghan Pages: 157 - 182 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Farhad Fakhar-Izadi, Mehdi Dehghan We present a high-order spectral element method (SEM) using modal or hierarchical basis for modeling of the two-dimensional linear and non-linear partial differential equations in complex geometry. The discretization is based on conforming spectral element in space and finite difference methods in time. Unlike the nodal SEM which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases, due to their orthogonal properties, enables us to obtain the elemental matrices efficiently in the linear problems. The difficulty of implementation of modal approximations for non-linear problems is treated in this paper by expanding the non-linear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT) algorithm. Utilization of Fourier interpolation on equidistant points in the FFT algorithm can be suitable at minimizing aliasing error. On the other hand, the polynomial order of approximations for non-linear terms can be sufficiently large to capture major variations and render the aliasing error so small. The proposed approach also leads to finding numerical solution of a non-linear differential equation through solving a system of linear algebraic equations. The non-constant Jacobian, as well as any non-polynomial coefficients and nonlinearities arisen in the entries of mass and stiffness matrices are expanded in terms of the Legendre polynomials by FFT algorithm and so they can be produced in the complex geometry accurately even in a tensor product form. Efficiency and exponential convergence properties of the approximation scheme are validated through several non-trivial examples in the numerical results section.

Authors:Bo Wang; Dong Liang Pages: 183 - 204 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Bo Wang, Dong Liang In this paper, we propose and analyze a finite difference method for the nonlinear Schrödinger equations on unbounded domain by using artificial boundary conditions. Two artificial boundary conditions are introduced to restrict the original Schrödinger equations on an unbounded domain into an initial–boundary value problem with a bounded domain. Then, a finite difference scheme for the reduced problem is proposed. The important feature of the proposed scheme is that an extrapolation operator is introduced to treat the nonlinear term while the scheme keeps unconditionally stable and does not introduce any oscillations at the artificial boundaries. The proposed scheme with the discrete artificial boundary conditions is rigorously analyzed to yield the unconditional stability and the scheme is also proved to be convergent. Numerical examples are given to show the performance of our scheme.

Authors:E. Keshavarz; Y. Ordokhani; M. Razzaghi Pages: 205 - 216 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): E. Keshavarz, Y. Ordokhani, M. Razzaghi This paper presents an efficient numerical method for solving the initial and boundary value problems of the Bratu-type. In the proposed method, the Taylor wavelets are introduced, for the first time. An operational matrix of integration is derived and is utilized to reduce the Bratu-type initial and boundary value problems to a system of algebraic equations. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed wavelets method. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique and a comparison is made with the existing results.

Authors:Winfried Auzinger; Jana Burkotová; Irena Rachůnková; Victor Wenin Pages: 217 - 229 Abstract: Publication date: June 2018 Source:Applied Numerical Mathematics, Volume 128 Author(s): Winfried Auzinger, Jana Burkotová, Irena Rachůnková, Victor Wenin For impulsive boundary value problems whose solutions encounter discontinuities (jumps) at a priori not known positions depending on the solution itself, numerical methods have not been considered so far. We extend the well-known shooting approach to this case, combining Newton iteration with the numerical solution of impulsive initial value problems. We discuss conditions necessary for the procedure to be well-defined, and we present numerical results for several examples obtained with an experimental code realized in MATLAB. 1

Authors:Peizhen Wang; Ming Sun; Changhui Yao Pages: 40 - 55 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Peizhen Wang, Ming Sun, Changhui Yao In this paper, the superconvergent property is found for the interpolation error of the nonconforming finite element at element centers. Based upon this property, the superconvergence results in the discrete l 2 norm for the solutions E → , H → and c u r l → E → are presented for the 3D time-harmonic Maxwell's equations. In order to get the global superconvergence, a new postprocess operator derived from the rotated Q 1 element interpolation is constructed, which is based on the superconvergence points. All theoretical results are justified by the provided smoothing and non-smoothing numerical tests.

Authors:D. Barrera; F. Elmokhtari; D. Sbibih Pages: 78 - 94 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): D. Barrera, F. Elmokhtari, D. Sbibih For solving a Fredholm integral equation of the second kind, we approximate its kernel by two types of bivariate spline quasi-interpolants, namely the tensor product and the continuous blending sum of univariate spline quasi-interpolants. We give the construction of the approximate solutions, and we prove some theoretical results related to the approximation errors of these methods. We illustrate the obtained results by some numerical tests giving a comparison with several methods in the literature.

Authors:H. Mahdi; A. Abdi; G. Hojjati Pages: 95 - 109 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): H. Mahdi, A. Abdi, G. Hojjati This paper concerns on the introduction of a method for solving a class of Volterra integro-differential equations (VIDEs) of the second kind. It is based on the combination of special general linear methods for ordinary differential equations and Gregory quadrature rule and equipped with a starting procedure. The convergence and stability of the method are analyzed. Some numerical experiments are given to illustrate the agreement of our implementation with the theoretical convergence orders and show the capability of the method in solving nonlinear VIDEs.

Authors:Stefano De Marchi; Giacomo Elefante Pages: 110 - 124 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Stefano De Marchi, Giacomo Elefante In this paper we consider two sets of points for Quasi-Monte Carlo integration on two-dimensional manifolds. The first is the set of mapped low-discrepancy sequence by a measure preserving map, from a rectangle U ⊂ R 2 to the manifold. The second is the greedy minimal Riesz s-energy points extracted from a suitable discretization of the manifold. Thanks to the Poppy-seed Bagel Theorem we know that the classes of points with minimal Riesz s-energy, under suitable assumptions, are asymptotically uniformly distributed with respect to the normalized Hausdorff measure. They can then be considered as quadrature points on manifolds via the Quasi-Monte Carlo (QMC) method. On the other hand, we do not know if the greedy minimal Riesz s-energy points are a good choice to integrate functions with the QMC method on manifolds. Through theoretical considerations, by showing some properties of these points and by numerical experiments, we attempt to answer to these questions.

Authors:Giovanna Califano; Dajana Conte Pages: 125 - 141 Abstract: Publication date: Available online 10 January 2018 Source:Applied Numerical Mathematics Author(s): Giovanna Califano, Dajana Conte We introduce domain decomposition methods of Schwarz waveform relaxation (WR) type for fractional diffusion-wave equations. We show that the Dirichlet transmission conditions among the subdomains lead to slow convergence. So, we construct optimal transmission conditions at the artificial interfaces and we prove that optimal Schwarz WR methods on N subdomains converge in N iterations both on infinite spatial domains and on finite spatial domains. We also propose optimal transmission conditions when the original problem is spatially discretized and we prove the same result found in the continuous case.

Authors:Yuan Li; Yanjie Ma; Rong An Pages: 142 - 163 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Yuan Li, Yanjie Ma, Rong An In this paper, we focus on a decoupled, linearized semi-implicit Galerkin FEM scheme for a MHD system coupled by the time-dependent Navier–Stokes equation with the steady Maxwell's equations in three-dimensional convex domain. First, additional regularities of the solution to the coupled MHD system are derived. By using H 1 -conforming finite element to approximate the magnetic field, it is shown that the proposed semi-linearized scheme is of the first-order convergence order of the velocity field, the magnetic field and the pressure under the time step condition Δ t = O ( h ) .

Authors:Jialin Hong; Lihai Ji; Zhihui Liu Pages: 164 - 178 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Jialin Hong, Lihai Ji, Zhihui Liu In this paper, we propose a conservative local discontinuous Galerkin method for a one-dimensional nonlinear Schrödinger equation. By using special generalized alternating numerical fluxes, we establish the optimal rate of convergence O ( h k + 1 ) , with polynomial of degree k and grid size h. Meanwhile, we show that this method preserves the charge conservation law. Numerical experiments verify our theoretical result.

Authors:Rodolfo Araya; Ramiro Rebolledo Pages: 179 - 195 Abstract: Publication date: Available online 4 January 2018 Source:Applied Numerical Mathematics Author(s): Rodolfo Araya, Ramiro Rebolledo In this work we develop an a posteriori error estimator, of the hierarchical type, for the Local Projection Stabilized (LPS) finite element method introduced in [5], applied to the incompressible Navier–Stokes equations. The technique use the solution of locals problems posed on appropriate finite dimensional spaces of bubble-like functions, to approach the error. Several numerical tests confirm the theoretical properties of the estimator and its performance.

Authors:Qing Ai; Hui-yuan Li; Zhong-qing Wang Pages: 196 - 210 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Qing Ai, Hui-yuan Li, Zhong-qing Wang Fully diagonalized spectral methods using Sobolev orthogonal/biorthogonal basis functions are proposed for solving second order elliptic boundary value problems. We first construct the Fourier-like Sobolev polynomials which are mutually orthogonal (resp. bi-orthogonal) with respect to the bilinear form of the symmetric (resp. unsymmetric) elliptic Neumann boundary value problems. The exact and approximation solutions are then expanded in an infinite and truncated series in the Sobolev orthogonal polynomials, respectively. An identity is also established for the a posterior error estimate with a simple error indicator. Further, the Fourier-like Sobolev orthogonal polynomials and the corresponding Legendre spectral method are proposed in parallel for Dirichlet boundary value problems. Numerical experiments illustrate that our Legendre methods proposed are not only efficient for solving elliptic problems but also equally applicable to indefinite Helmholtz equations and singular perturbation problems.

Authors:Feng-Nan Hwang; Yi-Zhen Su; Chien-Chou Yao Pages: 211 - 225 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Feng-Nan Hwang, Yi-Zhen Su, Chien-Chou Yao We develop and study a framework of multiscale finite element method (MsFEM) for solving the elliptic interface problems. Finding an appropriate boundary condition setting for local multiscale basis function problems is the current topic in the MsFEM research. In the proposed framework, which we call the iteratively adaptive MsFEM (i-ApMsFEM), the local-global information exchanges through iteratively updating the local boundary condition. Once the multiscale solution is recovered from the solution of global numerical formulation on coarse grids, which couples these multiscale basis functions, it provides feedback for updating the local boundary conditions on each coarse element. The key step of i-ApMsFEM is to perform a few smoothing iterations for the multiscale solution to eliminate the high-frequency error introduced by the inaccurate coarse solution before it is used for setting the boundary condition. As the method iterates, the quality of the MsFEM solution improves, since these adaptive basis functions are expected to capture the multiscale feature of the approximate solution more accurately. We demonstrate the advantage of the proposed method through some numerical examples for elliptic interface benchmark problems.

Authors:Rui M.P. Almeida; José C.M. Duque; Jorge Ferreira; Rui J. Robalo Pages: 226 - 248 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Rui M.P. Almeida, José C.M. Duque, Jorge Ferreira, Rui J. Robalo The aim of this paper is to study the convergence, properties and error bounds of the discrete solutions of a class of nonlinear systems of reaction–diffusion nonlocal type with moving boundaries, using the finite element method with polynomial approximations of any degree and some classical time integrators. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with a moving finite element method are investigated.

Authors:S. Chandra Sekhara Rao; Sheetal Chawla Pages: 249 - 265 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): S. Chandra Sekhara Rao, Sheetal Chawla In this work, we consider a parabolic system with an arbitrary number of linear singularly perturbed equations of reaction–diffusion type coupled in the reaction terms with a discontinuous source term. The diffusion term in each equation is multiplied by a small positive parameter, but these parameters may have different order of magnitude. The components of the solution have boundary and interior layers that overlap and interact. To obtain the approximate solution of the problem we construct a numerical method by combining the backward-Euler method on an uniform mesh in time direction, together with a central difference scheme on a variant of piecewise-uniform Shishkin mesh in space. We prove that the numerical method is uniformly convergent of first order in time and almost second order in spatial variable. Numerical experiments are presented to validate the theoretical results.

Authors:Xianju Yuan; Tianyu Tian; Hongni Zhou; Jiwei Zhou Pages: 266 - 279 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Xianju Yuan, Tianyu Tian, Hongni Zhou, Jiwei Zhou A highly accurate method for obtaining static deflections of a thin annular plate is helpful to effectively design the complicated structures with these plates. There have been numerous methods to achieve such a target. However, there is no direct technical literature for comparing these methods comprehensively. Therefore, the current study aims at performing comparison of three methods, optimization method (OM), finite element method (FEM), and harmonic differential quadrature (HDQ) method. Combining an instance, the comparisons give us insight into high accuracy and consistency of each other, showing high accuracy of the methods in this field. Compared with the results of FEM, the maximum error, less than 1%, demonstrates that the accuracy of the OM is high enough. Combining the small errors, the excellent stability of those brings a reliable method in this field. The maximum error and fluctuation drawn from the HDQ are evidently larger than those of the OM, and it is difficult to obtain results with higher accuracy based on the HDQ. Finally, the work described here suggests that the OM can be utilized to deal with such a complex problem in view of engineering and theory, and the HDQ method is more suitable to study the method for solving very complex partial differential systems of high order.

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.

Authors:Luigi Brugnano; Gianmarco Gurioli; Felice Iavernaro; Ewa B. Weinmüller Abstract: Publication date: Available online 24 December 2017 Source:Applied Numerical Mathematics Author(s): Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro, Ewa B. Weinmüller In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings.

Authors:Olga V. Ushakova Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): Olga V. Ushakova The aim of the paper is to give the numerical criteria for classification of different types of hexahedral cells which can emerge in a three-dimensional structured grid generation. In general, computational grids and their cells have to be nondegenerate, however, in practice, situations arise in which degenerate grids are used and computed. In these cases, to prevent lost of accuracy, special strategies must be chosen both in grid generation and physical phenomenon solution algorithms. To determine which cells need a modification in above strategies, degenerate cells have to be detected. The criteria are suggested for hexahedral cells constructed by a trilinear mapping of the unit cube. All hexahedral cells are divided into nondegenerate and degenerate. Among nondegenerate hexahedral cells, cells exotic in shape are singled out as inadmissible. Degenerate cells are divided into pyramids, prisms and tetrahedrons—types of cells which can be admissible in grid generation and solution algorithms. Inadmissible types of degenerations are also considered. An algorithm for testing three-dimensional structured grids according to suggested criteria is described. Both results of testing and examples of different types of cells are demonstrated. In conclusion, recommendations for structured grid generation with the purpose to exclude undesirable types of cells are given.

Authors:Hua Wang; Jinru Chen; Pengtao Sun; Fangfang Qin Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): Hua Wang, Jinru Chen, Pengtao Sun, Fangfang Qin A new conforming enriched finite element method is presented for elliptic interface problems with interface-unfitted meshes. The conforming enriched finite element space is constructed based on the P 1 -conforming finite element space. Approximation capability of the conforming enriched finite element space is analyzed. The standard conforming Galerkin method is considered without any penalty stabilization term. Our method does not limit the diffusion coefficient of the elliptic interface problem to a piecewise constant. Finite element errors in H 1 -norm and L 2 -norm are proved to be optimal. Numerical experiments are carried out to validate theoretical results.