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: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:I.V. Boykov; V.A. Roudnev; A.I. Boykova; O.A. Baulina Pages: 280 - 305 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): I.V. Boykov, V.A. Roudnev, A.I. Boykova, O.A. Baulina We propose a method for solving linear and nonlinear hypersingular integral equations. For nonlinear equations the advantage of the method is in rather weak requirements for the nonlinear operator behavior in the vicinity of the solution. The singularity of the kernel not only guarantees strong diagonal dominance of the discretized equations, but also guarantees the convergence of a simple iterative scheme based on Lyapunov stability theory. The resulting computational method can be implemented with recurrent neural networks or analog computers.
Authors:Gang Wu; Hong-Kui Pang; Jiang-Li Sun Pages: 306 - 323 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Gang Wu, Hong-Kui Pang, Jiang-Li Sun The approximation of exp ( t A ) B , where A is a large matrix and B a block vector, is a key ingredient in many scientific and engineering computations. A powerful tool to manage the matrix exponential function is to resort to a suitable rational approximation, such as the Carathéodory–Fejér approximation, whose core reduces to solving some shifted linear systems with multiple right-hand sides. However, these shifted systems are often difficult to solve when tA has a large norm. In this paper, we propose to solve some alternatively shifted linear systems. The motivation is that the magnitudes of the poles of the rational approximation are often medium-sized, and they can be much smaller than the norm of tA. We then introduce a shifted block FOM algorithm with deflated restarting for solving these alternatively shifted linear systems efficiently. Our method is advantageous when one has explicit access to A, and A − 1 can be computed directly. Theoretical results are given to show the rationale of the proposed strategy. The relationship between the approximations obtained from the shifted block FOM algorithm and the shifted block GMRES algorithm is also analyzed. Numerical experiments demonstrate the superiority of the proposed algorithm over many state-of-the-art algorithms for the matrix exponential.
Authors:Georges Chamoun; Mazen Saad; Raafat Talhouk Pages: 324 - 348 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Georges Chamoun, Mazen Saad, Raafat Talhouk This paper is devoted to the numerical study of a model arising from biology, consisting of chemotaxis equations coupled to Navier–Stokes flow through transport and external forcing. A detailed convergence analysis of this chemotaxis–fluid model by means of a suitable combination of the finite volume method and the nonconforming finite element method is investigated. In the case of nonpositive transmissibilities, a correction of the diffusive fluxes is necessary to maintain the monotonicity of the numerical scheme. Finally, many numerical tests are given to illustrate the behavior of the anisotropic Keller–Segel–Stokes system.
Authors:A. Abudawia; A. Mourad; J.H. Rodrigues; C. Rosier Pages: 349 - 369 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): A. Abudawia, A. Mourad, J.H. Rodrigues, C. Rosier We simulate a sharp-diffuse interface model issuing from a seawater intrusion problem in unconfined aquifer. We study a semi-implicite in time scheme for a P k , ( k ≥ 1 ) Lagrange finite element approximation. Using the specific regularity of the exact solution, we state that the scheme is of order 1 in time and k in space. We propose a finite volume method for a regular mesh and we compare the results given by these two approximations.
Authors:Linna Liu; Mengling Li; Feiqi Deng Pages: 370 - 386 Abstract: Publication date: May 2018 Source:Applied Numerical Mathematics, Volume 127 Author(s): Linna Liu, Mengling Li, Feiqi Deng In this paper, the equivalence theorem for the mean square exponential stability between the neutral delayed stochastic differential equations (NDSDEs) and the Euler–Maruyama numerical scheme is investigated via the continuous time Euler–Maruyama solutions. Firstly, with some preliminaries on basic notations and assumptions, we establish the approximation degree of the numerical scheme to the underlying NDSDE under the global Lipschitz condition for the dynamics and contractive mapping condition for the neutral operator of the equation, which guarantee the existence and uniqueness of the global solution. Then we show that the underlying NDSDE is exponentially stable in mean square if and only if, for some sufficiently small stepsize, the Euler–Maruyama numerical scheme is exponentially stable in mean square. With such a theoretical result, the mean square exponential stability of NDSDEs can be affirmed just by the simulation approach in practice. Finally, a constructive example is proposed to verify the theoretical result by simulation. Relatively, some analysis around the present topic will be given by remarks and some challenging problems for further works will be proposed in the conclusion section.
Authors:Mehdi Dehghan; Mostafa Abbaszadeh Pages: 92 - 112 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Mehdi Dehghan, Mostafa Abbaszadeh One meshless method based on the variational weak form is the element free Galerkin method. The element free Galerkin (EFG) method is similar to the finite element method but the test and trial functions for the EFG method are chosen from moving least squares (MLS) approximations. The shape functions of MLS approximation don't have the δ-Kronecker property thus the essential boundary conditions (Dirichlet boundary conditions) can not be applied, directly. On the other hand, there are some shape functions that have the mentioned property. One of these functions is the radial point interpolation method (RPIM). In the current paper, we employ the shape functions of RPIM as the test and trial functions. We apply the EFG method based on the RPIM (EFG-RPIM) for solving two-dimensional solute transport problems. To reduce the used CPU time, the proper orthogonal decomposition (POD) approach has been combined with the EFG-RPIM technique. Also, the unconditional stability and convergence of POD-EFG-RPIM method are proved by the energy method. Finally, some numerical results have been reported to show the efficiency and computational order of the new method.
Authors:Matthew J. Colbrook; Athanasisos S. Fokas Pages: 1 - 17 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Matthew J. Colbrook, Athanasisos S. Fokas Recently a new transform method, called the Unified Transform or the Fokas method, for solving boundary value problems (BVPs) for linear and integrable nonlinear partial differential equations (PDEs) has received a lot of attention. For linear elliptic PDEs, this method yields two equations, known as the global relations, coupling the Dirichlet and Neumann boundary values. These equations can be used in a collocation method to determine the Dirichlet to Neumann map. This involves expanding the unknown functions in terms of a suitable basis, and choosing a set of collocation points at which to evaluate the global relations. Here, using these methods for the Helmholtz and modified Helmholtz equations and following the earlier results of [15], we determine eigenvalues of the Laplacian in a convex polygon. Eigenvalues are characterised by the points where the generalised Dirichlet to Neumann map becomes singular. We find that the method yields spectral convergence for eigenfunctions smooth on the boundary and for non-smooth boundary values, the rate of convergence is determined by the rate of convergence of expansions in the chosen Legendre basis. Extensions to the case of oblique derivative boundary conditions and constant coefficient elliptic PDEs are also discussed and demonstrated.
Authors:Marzieh Dehghani-Madiseh; Milan Hladík Pages: 18 - 33 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Marzieh Dehghani-Madiseh, Milan Hladík We investigate the interval generalized Sylvester matrix equation A X B + C X D = F . We propose a necessary condition for its solutions, and also a sufficient condition for boundedness of the whole solution set. The main effort is performed to develop techniques for computing outer estimations of the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator, reducing significantly computational complexity, compared to the Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of A, B, C and D and in both of them we suppose that the midpoints of A and C are simultaneously diagonalizable as well as for the midpoints of the matrices B and D. Numerical experiments are given to illustrate the performance of the proposed methods.
Authors:Jinwei Fang; Boying Wu; Wenjie Liu Pages: 34 - 52 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Jinwei Fang, Boying Wu, Wenjie Liu In this paper, we present a stable and efficient numerical scheme for the linearized Korteweg–de Vries equation on unbounded domain. After employing the Crank–Nicolson method for temporal discretization, the transparent boundary conditions are derived for the time semi-discrete scheme. Then the unconditional stability of the resulting initial boundary problem is established. For spatial discretization, we construct a non-polynomial based spectral collocation method in which the basis functions are built upon a generalized Birkhoff interpolation. The interpolation error of the new basis is also investigated. Moreover, the basis functions build in two free parameters intrinsically which can be chosen properly so that the implicit time semi-discrete scheme collapses to an explicit scheme after spatial discretization. Numerical tests are performed to demonstrate the stability and accuracy of the proposed method.
Authors:Mikael Barboteu; Krzysztof Bartosz; David Danan Pages: 53 - 77 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Mikael Barboteu, Krzysztof Bartosz, David Danan We consider a dynamic process of frictional contact between a non-clamped viscoelastic body and a foundation. We assume that the normal contact response depends on the depth of penetration of the foundation by the considered body, and the dependence between these two quantities is governed by normal compliance conditions. On the other hand, the friction force is assumed to be a nonmonotone function of the slip rate where the friction threshold also depends on the depth of the penetration. Our aim in this paper is twofold. The first one is to prove the existence and the uniqueness of a weak solution for the contact problem under consideration. The second one is to provide the numerical analysis of the process involving its semi-discrete and fully discrete approximation as well as estimation of the error for both numerical schemes and the validation of such a result.
Authors:F. Dell'Accio; F. Di Tommaso; O. Nouisser; B. Zerroudi Pages: 78 - 91 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): F. Dell'Accio, F. Di Tommaso, O. Nouisser, B. Zerroudi In this paper we discuss an improvement of the triangular Shepard operator proposed by Little to extend the Shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and derivative data, has cubic approximation order and a good accuracy of approximation. Suggestions on how to avoid the use of derivative data, without losing both order and accuracy of approximation, are given.
Authors:Toby Sanders Pages: 1 - 9 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Toby Sanders Popular methods for finding regularized solutions to inverse problems include sparsity promoting ℓ 1 regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate the problem of the often debated selection of λ, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically argue for a scaling of the operators for a uniform parameter selection for all orders of HOTV regularization. In particular, parameter selection for all orders of HOTV may be determined by scaling an initial parameter for TV, which the imaging community may be more familiar with. We also provide several numerical results which justify our theoretical findings.
Authors:Xiaojun Tang; Heyong Xu Pages: 51 - 67 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Xiaojun Tang, Heyong Xu The main purpose of this work is to develop an integral pseudospectral scheme for solving integro-differential equations. We provide new pseudospectral integration matrices (PIMs) for the Legendre–Gauss and the flipped Legendre–Gauss–Radau points, respectively, and present an efficient and stable approach to computing the PIMs via the recursive calculation of Legendre integration matrices. Furthermore, we provide a rigorous convergence analysis for the proposed pseudospectral scheme in both L ∞ and L 2 spaces via a linear integral equation, and the spectral rate of convergence is demonstrated by numerical results.
Authors:Weiyang Ding; Michael Ng; Yimin Wei Pages: 68 - 85 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Weiyang Ding, Michael Ng, Yimin Wei In this paper, we study a fast algorithm for finding stationary joint probability distributions of sparse Markov chains or multilinear PageRank vectors which arise from data mining applications. In these applications, the main computational problem is to calculate and store solutions of many unknowns in joint probability distributions of sparse Markov chains. Our idea is to approximate large-scale solutions of such sparse Markov chains by two components: the sparsity component and the rank-one component. Here the non-zero locations in the sparsity component refer to important associations in the joint probability distribution and the rank-one component refers to a background value of the solution. We propose to determine solutions by formulating and solving sparse and rank-one optimization problems via closed form solutions. The convergence of the truncated power method is established. Numerical examples of multilinear PageRank vector calculation and second-order web-linkage analysis are presented to show the efficiency of the proposed method. It is shown that both computation and storage are significantly reduced by comparing with the traditional power method.
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.
Authors:V.C. Le; H.T. Pham; T.T.M. Ta Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): V.C. Le, H.T. Pham, T.T.M. Ta In the context of structural optimization in fluid mechanics we propose a numerical method based on a combination of the classical shape derivative and Hadamard's boundary variation method. Our approach regards the viscous flows governed by Stokes equations with the objective function of energy dissipation and a constrained volume. The shape derivative is computed by Lagrange's approach via the solutions of Stokes and adjoint systems. The programs are written in FreeFem++ using the Finite Element method.
Authors:Rekha P. Kulkarni; Gobinda Rakshit Abstract: Publication date: Available online 18 December 2017 Source:Applied Numerical Mathematics Author(s): Rekha P. Kulkarni, Gobinda Rakshit Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider discrete versions of the modified projection method and of the iterated modified projection method for solution of a Urysohn integral equation with a smooth kernel. For r ≥ 1 , a space of piecewise polynomials of degree ≤ r − 1 with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at r Gauss points. The orders of convergence which we obtain for these discrete versions indicate the choice of numerical quadrature which preserves the orders of convergence. Numerical results are given for a specific example.
Authors:Xinpeng Yuan; Chunguang Xiong Guoqing Zhu Abstract: Publication date: Available online 14 December 2017 Source:Applied Numerical Mathematics Author(s): Xinpeng Yuan, Chunguang Xiong, Guoqing Zhu In this paper, employing ideas developed for conservation law equations such as the Lax–Friedrich-type and Godunov-type numerical fluxes, we describe the numerical schemes for approximating the solution of the limit problem arising in the homogenization of Hamilton–Jacobi equations. All approximation methods involve three steps. The first scheme is a provably monotonic discretization of the cell problem for approximating the effective Hamiltonian for a given vector P ∈ R N . Next, using interpolation, we present an approximation of the effective Hamiltonian in the domain R N . Finally, the numerical schemes of the Hamilton–Jacobi equations with the effective Hamiltonian approximation are constructed. We also present global error estimates including all the discrete mesh sizes. The theoretical results are illustrated through numerical examples, including two convex Hamiltonians and two non-convex Hamiltonians.
Authors:D.D. Tcheutia; A.S. Jooste; W. Koepf Abstract: Publication date: Available online 14 November 2017 Source:Applied Numerical Mathematics Author(s): D.D. Tcheutia, A.S. Jooste, W. Koepf Using the q-version of Zeilberger's algorithm, we provide a procedure to find mixed recurrence equations satisfied by classical q-orthogonal polynomials with shifted parameters. These equations are used to investigate interlacing properties of zeros of sequences of q-orthogonal polynomials. In the cases where zeros do not interlace, we give some numerical examples to illustrate this.
Authors:Yongchao Yu; Jigen Peng Abstract: Publication date: Available online 3 November 2017 Source:Applied Numerical Mathematics Author(s): Yongchao Yu, Jigen Peng This paper studies algorithms for solving quadratically constrained ℓ 1 minimization and Dantzig selector which have recently been widely used to tackle sparse recovery problems in compressive sensing. The two optimization models can be reformulated via two indicator functions as special cases of a general convex composite model which minimizes the sum of two convex functions with one composed with a matrix operator. The general model can be transformed into a fixed-point problem for a nonlinear operator which is composed of a proximity operator and an expansive matrix operator, and then a new iterative scheme based on the expansive matrix splitting is proposed to find fixed-points of the nonlinear operator. We also give some mild conditions to guarantee that the iterative sequence generated by the scheme converges to a fixed-point of the nonlinear operator. Further, two specific proximal fixed-point algorithms based on the scheme are developed and then applied to quadratically constrained ℓ 1 minimization and Dantzig selector. Numerical results have demonstrated that the proposed algorithms are comparable to the state-of-the-art algorithms for recovering sparse signals with different sizes and dynamic ranges in terms of both accuracy and speed. In addition, we also extend the proposed algorithms to solve two harder constrained total-variation minimization problems.
Authors:Michael Weise Abstract: Publication date: Available online 31 October 2017 Source:Applied Numerical Mathematics Author(s): Michael Weise Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfürth for the related biharmonic equation. This article generalises Verfürth's result to Kirchhoff plates with an anisotropic material, which requires some modifications. Special emphasis is laid on the reduced Hsieh–Clough–Tocher triangular finite element, the conforming element with the least possible number of unknowns.
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