Abstract: Publication date: Available online 21 May 2019Source: Applied Numerical MathematicsAuthor(s): Susanne Beckers, Jörn Behrens, Winnifried Wollner Goal-oriented mesh adaptation, in particular using the dual-weighted residual (DWR) method, is known in many cases to produce very efficient meshes. For obtaining such meshes the (numerical) solution of an adjoint problem is needed to weight the residuals appropriately with respect to their relevance for the overall error. For hyperbolic problems already the weak primal problem requires in general an additional entropy condition to assert uniqueness of solutions; this difficulty is also reflected when considering adjoints to hyperbolic problems involving discontinuities where again an additional requirement (reversibility) is needed to select appropriate solutions.Within this article, an approach to the DWR method for hyperbolic problems based on an artificial viscosity approximation is proposed. It is discussed why the proposed method provides a well-posed dual problem, while a direct, formal, application of the dual problem does not. Moreover, we will discuss a further, novel, approach in which the forward problem need not be modified, thus allowing for an unchanged forward solution. The latter procedure introduces an additional residual term in the error estimation, accounting for the inconsistency between primal and dual problem.Finally, the effectivity of the extended error estimator, assessing the global error by a suitable functional of interest, is tested numerically; and the advantage over a formal estimator approach is demonstrated.

Abstract: Publication date: Available online 21 May 2019Source: Applied Numerical MathematicsAuthor(s): Mair Khan, Arif Hussain, M.Y. Malik, T. Salahuddin, Shaban Aly The flows of fluid due to an upper horizontal surface of paraboloid of revolution have great significance in the field of science and aerodynamic industry such as surface of a rocket, bonnet of a car and a pointed surface of an aircraft etc. Hence, present analysis focuses on the study of bioconvection MHD Carreau nanofluid flow over a paraboloid surface of revolution. Cattaneo-Christov model has been constructed by using universal Fourier's and Fick's laws to explore the heat and mass flux phenomena. The governing PDE's are transformed into ODE's via similarity approach and then numerically solved by shooting technique. The effects of relevant non dimensional governing physical parameters on velocity, energy and concentration profiles along with wall friction factor, wall heat and mass transfer coefficients are deliberated with help of graphs and tables. The calculated results for heat and mass diffusion (using generalized Fourier's and Fick's laws) are better than classical Fourier's and Fick's laws. Thus, the main aim of current paper is to calculate the unique results for generalized Fourier's and Fick's laws on nanoscale.

Abstract: Publication date: October 2019Source: Applied Numerical Mathematics, Volume 144Author(s): F. Ghanbari, P. Mokhtary, K. Ghanbari This paper focuses on providing a novel high-order algorithm for the numerical solution of a class of linear semi-explicit fractional order integro-differential algebraic equations with weakly singular kernels using the discrete Legendre collocation method. For this purpose, we investigate the existence and uniqueness as well as the regularity properties of this problem and show that some derivatives of its solutions suffer from discontinuity at the left endpoint of the integration domain dependence on both fractional derivative order and the weakly singular kernel function. To remove the singularity, using a new smoothing transformation the main equation is converted into an equivalent equation with better regularity properties and the Legendre collocation discretization method is designed for the transformed equation. The convergence properties of the proposed method are also studied, and we show that the new strategy exhibits a high-order of convergence even when the exact solutions are non-smooth. Some numerical examples are given to illustrate the performance of the approach.

Abstract: Publication date: Available online 16 May 2019Source: Applied Numerical MathematicsAuthor(s): Fahimeh Saberi Zafarghandi, Maryam Mohammadi The radial basis functions collocation method is developed to solve Riesz space fractional advection-dispersion equation (RSFADE). To do this, we first provide the Riemann-Liouville fractional derivatives of the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. Then a method of lines, implemented as a meshless method based on spatial trial spaces spanned by the RBFs is developed for the numerical solution of the RSFADE. Numerical experiments are presented to validate the newly developed method and to investigate accuracy and efficiency. The numerical rate of convergence in space is computed. The stability of the linear systems arising from discretizing the Riesz fractional derivative with RBFs is also analysed.

Abstract: Publication date: Available online 15 May 2019Source: Applied Numerical MathematicsAuthor(s): Yanping Chen, Jinling Zhang, Yunqing Huang, Yeqing Xu In this paper, we investigate the hp spectral element approximation for optimal control problem governed by elliptic equation with an integral constraint for state. We first present the optimality conditions of the continuous and discretized control problems, respectively. Then, hp a posteriori error estimates are obtained for the coupled state and control approximation. In the end, we use a simple and yet efficient projection algorithm to carry out ample numerical experiments which confirm our analytical results.

Abstract: Publication date: Available online 14 May 2019Source: Applied Numerical MathematicsAuthor(s): Muhammed Awwal Aliyu, Poom Kumam, Bala Abubakar Auwal In this paper, a modified Hestenes-Stiefel (HS) spectral conjugate gradient (CG) method for monotone nonlinear equations with convex constraints is proposed based on projection technique. The method can be viewed as an extension of a modified HS-CG method for unconstrained optimization proposed by Amini et al. (Optimization Methods and Software, pp: 1-13, 2018). A new search direction is obtained by incorporating the idea of spectral gradient parameter and some modification of the conjugate gradient parameter. The proposed method is derivative-free and requires low memory which makes it suitable for large scale monotone nonlinear equations. Global convergence of the method is established under suitable assumptions. Preliminary numerical comparisons with some existing methods are given to show the efficiency of our proposed method. Furthermore, the proposed method is successfully applied to solve sparse signal reconstruction in compressive sensing.

Abstract: Publication date: Available online 13 May 2019Source: Applied Numerical MathematicsAuthor(s): Jiyong Li, Xinyuan Wu Continuous-stage extended Runge-Kutta-Nyström (CSERKN) methods are proposed and developed for oscillatory problem q″(t)+Mq(t)=f(q(t)). These new methods take into account the special feature of the oscillatory problem so that they integrate exactly unperturbed problem q″(t)+Mq(t)=0. When this problem can be regarded as a Hamiltonian system, we show sufficient conditions for energy-preservation in terms of the coefficients of the method. We also study the symmetry and stability of the methods. Two symmetric and energy-preserving CSERKN schemes of order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.

Abstract: Publication date: Available online 10 May 2019Source: Applied Numerical MathematicsAuthor(s): Mahmoud A. Zaky Tempered fractional-order models open up new possibilities for robust mathematical modeling of complex multi-scale problems and anomalous transport phenomena. The purpose of this paper is twofold. First, we study existence, uniqueness, and structural stability of solutions to nonlinear tempered fractional differential equations involving the Caputo tempered fractional derivative with generalized boundary conditions. Second, we develop and analyze a singularity preserving spectral-collocation method for the numerical solution of such equations. We derive rigorous error estimates under the Lωθ−1,02- and L∞-norms. The most remarkable feature of the method is its capability to achieve spectral convergence for the solution with limited regularity. The results confirm that the method is best suited to discretize tempered fractional differential equations as they naturally take the singular behavior of the solution into account.

Abstract: Publication date: Available online 10 May 2019Source: Applied Numerical MathematicsAuthor(s): Mostafa Abbaszadeh, Mehdi Dehghan A delay PDE is different from a PDE in which it depends not only on the solution at a present stage but also on the solution at some past stage(s). In the current paper, we develop interpolating stabilized EFG method for a neutral delay PDE with fractional derivative in terms of Caputo fractional derivatives. The considered model in this paper is a generalized form of the other equations such that it contains a delay term. Thus, the existence and uniqueness of its analytic solution have been proven by the separation variable technique. At first, the temporal direction is discretized with a finite difference scheme. Then the EFG method has been employed to discrete the spatial direction. The stability of time-discrete scheme is studied by using the energy method. Also, we show the convergence order of the developed technique is O(τ3−α) in the temporal direction and the convergence order in spatial direction is O(rp+1). Finally, two test problems have been considered to demonstrate the capability and efficiency of the proposed numerical technique.

Abstract: Publication date: Available online 9 May 2019Source: Applied Numerical MathematicsAuthor(s): Meng Zhao, Hong Wang Fractional partial differential equations (FPDEs) provide very competitive tools to model challenging phenomena involving anomalous diffusion or long-range memory and spatial interactions. However, numerical methods for space-time FPDEs generate dense stiffness matrices and involve numerical solutions at all the previous time steps, and so have O(N2+MN) memory requirement and O(MN3+M2N) computational complexity where N and M are the numbers of spatial unknowns and time steps, respectively.We develop and analyze a finite difference method (FDM) for space-time FPDEs in three space dimensions with a combination of Dirichlet and fractional Neumann boundary conditions, in which a shifted Grünwald discretization is used in space and an L1 discretization is used in time so the derived FDM has virtually first-order accuracy. We then derive different fast FDMs by carefully analyzing the structure of the stiffness matrix of numerical discretization as well as the coupling in the time direction. The resulting fast FDMs have an almost linear computational complexity and linear storage with respect to the number of spatial unknowns as well as time steps. Numerical experiments are presented to demonstrate the utility of the methods.

Abstract: Publication date: Available online 8 May 2019Source: Applied Numerical MathematicsAuthor(s): Mahboub Baccouch In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

Abstract: Publication date: Available online 8 May 2019Source: Applied Numerical MathematicsAuthor(s): Kim Yunho In this work, we interpret real symmetric eigenvalue problems in an unconstrained global optimization framework. More precisely, given two N×N matrices, a symmetric matrix A, and a symmetric positive definite matrix B, we propose and analyze a nonconvex functional F whose local minimizers are, indeed, global minimizers. These minimizers correspond to eigenvectors of the generalized eigenvalue problem Ax=λBx associated with its smallest eigenvalue. To minimize the proposed functional F, we consider the gradient descent method and show its global convergence. Furthermore, we provide explicit error estimates for eigenvalues and eigenvectors at the kth iteration of the method in terms of the gradient of F at the kth iterate xk. At the end, we provide a few numerical experiments to confirm our analysis and to compare with other methods, which reveals interesting numerical aspects of our proposed model.

Abstract: Publication date: Available online 7 May 2019Source: Applied Numerical MathematicsAuthor(s): P. Junghanns, R. Kaiser We discuss the problem of the instability of the numerical method proposed by Kalandiya [10] for the approximate solution of the two-dimensional elasticity problem of a crack at a circular cavity surface. Furthermore, we describe a method to overcome this problem.

Abstract: Publication date: Available online 7 May 2019Source: Applied Numerical MathematicsAuthor(s): Hiba Fareed, John R. Singler In our earlier work Fareed et al. (2018) [1], we developed an incremental approach to compute the proper orthogonal decomposition (POD) of PDE simulation data. Specifically, we developed an incremental algorithm for the SVD with respect to a weighted inner product for the discrete time POD computations. For continuous time data, we used an approximate approach to arrive at a discrete time POD problem and then applied the incremental SVD algorithm. In this note, we analyze the continuous time case with simulation data that is piecewise constant in time such that each data snapshot is expanded in a finite collection of basis elements of a Hilbert space. We first show that the POD is determined by the SVD of two different data matrices with respect to weighted inner products. Next, we develop incremental algorithms for approximating the two matrix SVDs with respect to the different weighted inner products. Finally, we show neither approximate SVD is more accurate than the other; specifically, we show the incremental algorithms return equivalent results.

Abstract: Publication date: Available online 7 May 2019Source: Applied Numerical MathematicsAuthor(s): Jing Sun, Daxin Nie, Weihua Deng Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of fast algorithms.

Abstract: Publication date: Available online 7 May 2019Source: Applied Numerical MathematicsAuthor(s): Pradip Roul, Ravi Agarwal In the present work, the authors describe two B-spline collocation methods for a class of nonlinear singular Lane-Emden type equations which describe several phenomena in theoretical physics and astrophysics. The first method is based on non-optimal quintic B-spline collocation approach, while the second method is based on optimal quintic B-spline collocation approach. We note that the former method yields fourth-order convergent approximation to the solution of the second-order boundary value problems. To obtain an optimal convergence of order six to the solution of the same problem, the later method is designed by perturbing the original problem. Convergence results for both methods are established via Green's function approach. Four nonlinear examples are provided to illustrate the suggested methods and to verify their theoretical rates of convergence. It is observed that the first method yields non-optimal fourth order accuracy and the second method exhibits optimal sixth order accuracy for the solution of the problem. Moreover, numerical results appear to be higher accurate when compared to other methods.

Abstract: Publication date: Available online 4 May 2019Source: Applied Numerical MathematicsAuthor(s): Mohammad Hossein Heydari This study deals with a computational scheme based on the Chebyshev cardinal wavelets for a new class of nonlinear variable-order (V-O) fractional quadratic integral equations (QIEs). Through the way, a new operational matrix (OM) of V-O fractional integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal wavelets including undetermined coefficients. Thereafter, a system of nonlinear algebraic equations is extracted by substituting the mentioned expansion in the intended problem, utilizing the generated OM and considering the cardinal property of the basis functions. Finally, the approximate solution is obtained by solving the yielded system. Meanwhile, the convergence of the established approach is investigated in the Sobolev space. Moreover, the applicability and the accuracy of the method are examined by solving several numerical examples.

Abstract: Publication date: Available online 3 May 2019Source: Applied Numerical MathematicsAuthor(s): Jingjun Zhao, Rui Zhan, Yang Xu This paper is concerned with the convergence property of the Strang splitting for the Gardner equation. We assume that the Gardner equation is locally well-posed and the solution is bounded. We first obtain the regularity properties of the nonlinear divided equation. With these regularity properties, the Strang splitting is proved to converge at the expected rate in L2. Numerical experiments demonstrate the theoretical result and serve to compare the accuracy and efficiency of different time stepping methods. Finally, the proposed method is applied to simulate the multi solitons collisions for the Gardner equation.

Abstract: Publication date: Available online 2 May 2019Source: Applied Numerical MathematicsAuthor(s): Yanping Chen, Yang Wang, Yunqing Huang, Longxia Fu In this paper, we propose two efficient two-grid algorithms for the nonlinear parabolic equations by using expanded mixed finite element methods. To linearize the finite element equations, two-grid algorithms based on some Newton iteration approach and correction method are applied. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/3) (or H=O(h1/4)). Experiments are presented to show the efficiency and effectiveness of the two-grid methods.

Abstract: Publication date: Available online 2 May 2019Source: Applied Numerical MathematicsAuthor(s): Azadeh Ghasemifard, Mahdieh Tahmasebi In this work, we will show strong convergence of the Multilevel Monte-Carlo (MLMC) algorithm with split-step backward Euler (SSBE) and backward (drift-implicit) Euler (BE) schemes for nonlinear jump-diffusion stochastic differential equations (SDEs) when the drift coefficient is globally one-sided Lipschitz and the test function is only locally Lipschitz. We also confirm these theoretical results by numerical experiments for the jump-diffusion processes.

Abstract: Publication date: Available online 30 April 2019Source: Applied Numerical MathematicsAuthor(s): Sushil Kumar, Cécile Piret In this study, we propose a numerical discretization of space-time fractional partial differential equations (PDEs) with variable coefficients, based on the radial basis functions (RBF) and pseudospectral (PS) methods. The RBF method is used for space discretization, while Chebyshev polynomials handle time discretization. The use of PS methods significantly reduces the number of nodes needed to obtain the solution. The proposed numerical scheme is capable of handling all three types of boundary conditions: Dirichlet, Neumann and Robin. We give numerical examples to validate our method and to show its superior performance compared to other techniques.

Abstract: Publication date: Available online 30 April 2019Source: Applied Numerical MathematicsAuthor(s): Pouria Assari, Mehdi Dehghan The current investigation studies a numerical method to solve Volterra integral equations of the second kind arising in the single term fractional differential equations with initial conditions. The proposed method estimates the solution of the mentioned Volterra integral equations using the discrete Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least squares polynomial fitting. We compute fractional integrals appeared in the method by a suitable integration rule based on the non-uniform composite Gauss-Legendre quadrature formula. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve fractional differential equations and its algorithm can be easily implemented on computers. The error bound and the convergence rate of the presented method are obtained. Finally, numerical examples are included to show the validity and efficiency of the new technique.

Abstract: Publication date: Available online 26 April 2019Source: Applied Numerical MathematicsAuthor(s): E.V. Chistyakova, V.F. Chistyakov This paper addresses systems of linear integral differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Such systems can be viewed as differential algebraic equations perturbed by the Fredholm operators. For such problems, we obtain solvability conditions and discuss the influence of small perturbations of the input data on the solution. Within the existence theorems, we propose the least squares method to obtain a numerical approximation.

Abstract: Publication date: Available online 24 April 2019Source: Applied Numerical MathematicsAuthor(s): Zhendong Gu Spectral collocation methods are developed for weakly singular Volterra integro-differential equations (VIDEs). Convergence analysis results show that global convergence order depends on the regularities of the kernels functions and solutions to VIDEs, and the number of collocation points is independent of regularities of given functions and the solution to VIDEs. Numerical experiments are carried out to confirm these theoretical results. In numerical experiments, we develop a numerical scheme for nonlinear VIDEs, and investigate the local convergence on collocation points.

Abstract: Publication date: Available online 24 April 2019Source: Applied Numerical MathematicsAuthor(s): Christopher T.H. Baker, Neville J. Ford The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If a,b,c and {τˇℓ} are real and γ♮ is real-valued and continuous, an example with these parameters is(⋆)u′(t)={au(t)+bu(t+τˇ1)+cu(t+τˇ2)}+∫τˇ3τˇ4γ♮(s)u(t+s)ds. A wide class of equations (⋆), or of similar type, can be written in the “canonical” form(⋆⋆)u′(t)=∫τminτmaxu(t+s)dσ(s)(t∈R), for a suitable choice of τmin,τmax where σ is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (⋆⋆), there is a corresponding characteristic function(⋆⋆⋆)χ(ζ)):=ζ−∫τminτmaxexp(ζs)dσ(s)(ζ∈

Abstract: Publication date: Available online 19 April 2019Source: Applied Numerical MathematicsAuthor(s): Lu Zhang, Zhaojie Zhou In this paper we investigate spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation with control integral constraint. First order optimality condition is derived and the regularity of control problem is discussed. Based on first discretize then optimize approach a spectral Galerkin approximation of the control problem is developed, where two-sided Jacobi polyfractonomials are used to approximate the state variable and variational discretization is used to discretize the control variable. A priori error analysis for state variable, adjoint state variable and control variable is presented. Numerical experiments are carried out to illustrate the theoretical findings.

Abstract: Publication date: Available online 19 April 2019Source: Applied Numerical MathematicsAuthor(s): Mohammad Maleki, Ali Davari In this paper, we consider the nonlinear fractional retarded differential equations (FRDE). We extend the results of the existence and uniqueness of the solution, the propagation of derivative discontinuities and the dependence of the solution on the parameters of the equation. Next, we develop an efficient multistep collocation method for solving this type of equations. The proposed scheme is especially suited for FRDEs with piecewise smooth solutions, due to its essential feature of local approximations on subintervals. The stability of the scheme is accessed, and the convergence analysis is studied for functions in appropriate Sobolev spaces. Numerical results confirm the spectral accuracy and the stability of the proposed method for large domain calculations.

Abstract: Publication date: Available online 18 April 2019Source: Applied Numerical MathematicsAuthor(s): Yong Liu, Chuan-Qing Gu The variants of randomized Kaczmarz (RK) and randomized Gauss-Seidel (RGS) are distinct iterative algorithms for ridge regression. Theoretical convergence rates for these two algorithms were provided by a simple side-by-side analysis in recent work. Considering that greedy randomized Kaczmarz (GRK) algorithm converges faster than the RK algorithm in both theory and experiments, we extend the GRK algorithm for ordinary least squares (OLS) regression to ridge regression and establish the corresponding convergence theory for the resulting algorithm and, furthermore, we study the GRK algorithm with relaxation for ridge regression and prove that it also converges with expected exponential rate in this paper. In addition, we propose an accelerated GRK algorithm with relaxation for ridge regression by executing more rows that corresponding to the larger entries of the residual vector simultaneously at each iteration. Numerical results show that the GRK algorithm with and without relaxation for ridge regression perform much better in iteration steps than the variants of RK and RGS algorithms, and the accelerated GRK algorithm with relaxation significantly outperforms all other algorithms in terms of both iteration counts and computing times.

Abstract: Publication date: Available online 13 April 2019Source: Applied Numerical MathematicsAuthor(s): Maryam Bahmanpour, Majid Tavassoli Kajani, Mohammad Maleki The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system {xλ1,xλ2,…} with respect to the weight function w(x)=1 on [0,1]. In this paper, we introduce Müntz wavelets by using the Müntz-Legendre polynomials on the interval [0,1]. Using Jacobi polynomials we make the numerical evaluation of Müntz wavelets more stable. Next, this basis in combination with a matrix method is utilized to solve Fredholm integral equations of the first kind, which have many applications in several fields of computational physics. Errors of the proposed method are studied and numerical results are given to demonstrate the spectral accuracy of the method. We will show that the proposed method, in contrast to other wavelet methods, is capable of providing highly accurate results for solutions with fractional powers.

Abstract: Publication date: Available online 12 April 2019Source: Applied Numerical MathematicsAuthor(s): Jingyong Tang, Chengdai Huang, Yongli Wang In this paper we consider the symmetric cone complementarity problem with Cartesian P0-property (denoted by P0-SCCP) which includes the well-known monotone symmetric cone complementarity problem. We propose a predictor-corrector inexact smoothing algorithm for solving the P0-SCCP and prove that the method is globally and locally quadratically convergent under suitable assumptions. Especially, we prove that our algorithm can generate a bounded iteration sequence when the solution set of the P0-SCCP is nonempty and bounded, or the solution set of the monotone SCCP is nonempty. Moreover, the proposed algorithm solves the linear systems in both predictor step and corrector step only approximately by using an inexact Newton method. Hence when one solves large-scale SCCPs, our algorithm can save much computation work compared to existing smoothing-type algorithms. Numerical results confirm these good theoretical properties.

Abstract: Publication date: Available online 12 April 2019Source: Applied Numerical MathematicsAuthor(s): Hailong Qiu, Liquan Mei In this article, two multi-level stabilized algorithms for the Navier-Stokes equations with damping are studied. The algorithms combine the stabilized finite element technique with the multi-level method. Two kinds of multi-level stabilized algorithms based on the Oseen-type correction and Newton-type correction are presented. Some error estimates for the multi-level stabilized algorithms are derived. Moreover, the numerical tests agree with the theoretical results.

Abstract: Publication date: Available online 12 April 2019Source: Applied Numerical MathematicsAuthor(s): Lorenzo Mascotto, Alexander Pichler We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces with special functions capturing the physical behaviour of the solution to the target problem. As these two issues are directly related to an increase of the number of degrees of freedom, we use a reduction strategy inspired by [31], which allows to mitigate the growth of the dimension of the approximation space when considering h- and p-refinements. This renders the new method highly competitive in comparison to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz problem with piecewise constant wave number. A wide range of numerical experiments, including the p-version with quasi-uniform meshes and the hp-version with isotropic and anisotropic mesh refinements, is presented.

Abstract: Publication date: Available online 9 April 2019Source: Applied Numerical MathematicsAuthor(s): Weiping Shen, Yaohua Hu, Chong Li, Jen-Chih Yao In this paper, we study the convergence of the Newton-type methods for solving the square inverse singular value problem with possible multiple and zero singular values. Comparing with other known results, positivity assumption of the given singular values is removed. Under the nonsingularity assumption in terms of the (relative) generalized Jacobian matrices, quadratic/suplinear convergence properties (in the root-convergence sense) are proved. Moreover, numerical experiments are given in the last section to demonstrate our theoretic results.

Abstract: Publication date: Available online 8 April 2019Source: Applied Numerical MathematicsAuthor(s): Juan Chen, Fangqi Chen In this paper, a conservative and linearly implicit finite difference scheme with second order temporal accuracy and eighth order spatial accuracy by means of the Richardson extrapolation method is proposed for approximating solution to the initial-boundary problem for the Klein-Gordon-Schrödinger equations. Furthermore, the difference solution is shown to be bounded by taking advantage of the fact that the proposed scheme possesses two conservation laws, and with the aid of the discrete energy method, the difference scheme is demonstrated to be convergent in the maximum norm. Finally, numerical experiments are assigned to support the theoretical results.

Abstract: Publication date: Available online 5 April 2019Source: Applied Numerical MathematicsAuthor(s): Jikun Zhao, Bei Zhang, Xiaopeng Zhu The nonconforming virtual element method for the parabolic problem is developed in this paper, where the semi-discrete and fully discrete formulations are presented. In order to analyze the convergence of the method, we construct a projection operator in the sense of the energy norm and give the corresponding error estimates in the L2 norm and broken H1 semi-norm. With the help of the energy projection operator, we prove the optimal convergence of the nonconforming virtual element method in the semi-discrete and fully discrete formulations, respectively. Finally, we investigate the convergence of the nonconforming virtual element method by some numerical examples, which verify the theoretical results.

Abstract: Publication date: Available online 5 April 2019Source: Applied Numerical MathematicsAuthor(s): Pengfei Wang, Pengzhan Huang In this paper, the Crank-Nicolson extrapolation scheme is proposed for the Korteweg-de Vries equation, where a finite element method is applied for the spatial approximation, the time discretization is based on the Crank-Nicolson scheme for the linear term and the semi-implicit extrapolation scheme for the nonlinear term. Moreover, in numerical analysis, we split the error function into two parts, one from the spatial discretization and other one from the temporal discretization, by introducing a corresponding time-discrete system. Then, based on some regularity assumptions, we present optimal error estimate and prove that the scheme is almost unconditionally convergent, i.e., the scheme is convergent when the time step is less than or equal to a constant. Finally, numerical tests confirm the theoretical results of the presented method.

Abstract: Publication date: Available online 3 April 2019Source: Applied Numerical MathematicsAuthor(s): F. Dassi, G. Vacca We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier-Stokes problems and we provide a series of examples to numerically verify the theoretical behavior of high-order VEM.

Abstract: Publication date: Available online 3 April 2019Source: Applied Numerical MathematicsAuthor(s): Quanxiang Wang, Zhiyue Zhang In this paper, we propose a stabilized immersed finite volume element method for solving elliptic interface problems on Cartesian mesh. To improve the classic immersed finite volume element schemes, certain stability terms over interface edges are added. The new schemes presented here have the same global degrees of freedom as their classic counterparts. Optimal error estimates are derived in an energy norm under piecewise H2 regularity. Numerical examples demonstrate that the stabilized immersed finite volume element schemes are second order accuracy in L2 norm and the oscillating behavior of L∞ error can be improved.

Abstract: Publication date: Available online 3 April 2019Source: Applied Numerical MathematicsAuthor(s): Jianqiang Xie, Zhiyue Zhang, Dong Liang In this paper, an efficient energy-preserving splitting Crank-Nicolson difference scheme for the fractional-in-space Boussinesq equation is established. The novelty of the proposed scheme here is that the potential function v is introduced via ∂u∂t=−(−Δ)α2v to ensure the conservation of energy. By utilizing the discrete energy method, it is shown that the proposed scheme attains the convergence rates of O(Δt2+h2) in the discrete L∞- norm without any restrictions on the grid ratio, and is unconditionally stable. Finally, a linearized iterative algorithm is proposed and numerical results are provided to demonstrate the correctness of the theoretical results.

Abstract: Publication date: Available online 2 April 2019Source: Applied Numerical MathematicsAuthor(s): W.A. Gunarathna, H.M. Nasir, W.B. Daundasekera An explicit form for coefficients of shifted Grünwald type approximations of fractional derivatives is presented. This form directly gives approximations for any order of accuracy with any desired shift leading to efficient and automatic computations of the coefficients. To achieve this, we consider generating functions in the form of power of a polynomial. Then, an equivalent characterization for consistency and order of accuracy established on a general generating function is used to form a linear system of equations with Vandermonde matrix. This linear system is solved for the coefficients of the polynomial in the generating function. These generating functions completely characterize Grünwald type approximations with shifts and order of accuracy. Incidentally, the constructed generating functions happen to be a generalization of the previously known Lubich forms of generating functions without shift. We also present a formula to compute leading and some successive error terms from the coefficients. We further show that finite difference formulas for integer order derivatives with desired shift and order of accuracy are some special cases of our explicit form.

Abstract: Publication date: Available online 29 March 2019Source: Applied Numerical MathematicsAuthor(s): Matthew Kaplan, David P. Nicholls The scattering of linear electromagnetic waves by a layered structure is a central model in many problems of engineering interest. In this contribution, we focus on the interaction of visible and near–visible radiation with periodic structures on the micron or nanometer scales which are relevant for many applications in nanoplasmonics. The fabrication of such structures is extremely difficult and costly, and even the most sophisticated laboratories have difficulty measuring the dimensions of gratings they have constructed. We present a new, extremely rapid and robust, identification algorithm for providing precisely this information. It is built upon a Nonlinear Least Squares framework, and implemented with a High–Order Perturbation of Surfaces methodology which is orders of magnitude faster than volumetric solvers, while outperforming surface methods (such as Boundary Integral Methods) for the geometries we consider here. In addition to a full derivation and specification of the algorithm, we also support our claims with a number of illustrative simulations.

Abstract: Publication date: Available online 29 March 2019Source: Applied Numerical MathematicsAuthor(s): Jinghua Xie, Lijun Yi We propose and analyze an h-p version of the continuous Petrov-Galerkin time stepping method for nonlinear second-order delay differential equations with vanishing delays. We establish a priori error estimates in the L2- and L∞-norm that are completely explicit in the local time steps, in the local polynomial degrees, and in the local regularity of the exact solutions. In particular, it is proved that, for analytic solutions with start-up singularities exponential rates of convergence can be achieved by using geometrically refined time steps and linearly increasing approximation orders. Numerical examples are given illustrating the theoretical results.

Abstract: Publication date: Available online 26 March 2019Source: Applied Numerical MathematicsAuthor(s): Tamal Pramanick, Rajen Kumar Sinha In this exposition we study two-scale composite finite element approximations of parabolic problems with measure data in time for both convex and nonconvex polygonal domains. This research is motivated by the work of Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] on the composite finite element approximations of elliptic boundary value problems. The main features of the composite finite element method is that, it not only uses minimal dimension of the approximation space but also handle the domain boundary in a flexible and systematic manner, which is very advantageous for domains with complicated geometry. Both spatially semidiscrete and fully discrete approximations of the proposed method are analyzed. In the case of convex domains, we derive error estimate of order O(HLog˜1/2(H/h)+k1/2) in the L2(0,T;L2(Ω))-norm, where H and h denote the coarse-scale and fine-scale mesh size, respectively, and k is the time step. Further, an error estimate of order O(HsLog˜s/2(H/h)+k1/2), 1/2≤s≤1 is shown to hold in the L2(0,T;L2(Ω))-norm for nonconvex domains. Numerical experiment confirms the theoretical findings and reveals the potential of the composite finite element method.

Abstract: Publication date: Available online 5 December 2018Source: Applied Numerical MathematicsAuthor(s): Florian Monteghetti, Denis Matignon, Estelle Piot This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss-Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.

Abstract: Publication date: Available online 22 November 2018Source: Applied Numerical MathematicsAuthor(s): G.M. Coclite, A. Fanizzi, L. Lopez, F. Maddalena, S.F. Pellegrino In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering because seems to provide an effective approach to modeling mechanical systems avoiding spatial discontinuous derivatives and body singularities. In particular, we will consider the linear model of peridynamics in a one-dimensional spatial domain. Here we will review some numerical techniques to solve this equation and propose some new computational methods of higher order in space; moreover we will see how to apply the methods studied for the linear model to the nonlinear one. Also a spectral method for the spatial discretization of the linear problem will be discussed. Several numerical tests will be given in order to validate our results.