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Foreword by the Editor-in-Chief Chi-Wang Shu Communications on Applied Mathematics and Computation    2019, 1 (1): 1-1.   DOI: 10.1007/s42967-019-0010-2 Abstract （3422）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0）

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Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy Mark Ainsworth, Zhiping Mao Communications on Applied Mathematics and Computation    2019, 1 (1): 5-19.   DOI: 10.1007/s42967-019-0008-9 Abstract （817）      PDF       Knowledge map    Save We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution. The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy. We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case. In particular, we show that, analogous to the Cahn-Hilliard case, the solutions consist of regions in which the solution is a piecewise constant (whose value depends on the mass and the fractional order) separated by an interface whose width is independent of the mass and the fractional derivative. However, if the average value of the initial data exceeds some threshold (which we determine explicitly), then the solution will tend to a single constant steady state. Reference | Related Articles | Metrics | Comments（0）

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A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions Zachary Grant, Sigal Gottlieb, David C. Seal Communications on Applied Mathematics and Computation    2019, 1 (1): 21-59.   DOI: 10.1007/s42967-019-0001-3 Abstract （779）      PDF       Knowledge map    Save High-order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufcient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p=6, and defne an optimization procedure that allows us to fnd such SSP methods. Several types of these methods are presented and their efciency compared. Finally, these methods are tested on several PDEs to demonstrate the beneft of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases. Reference | Related Articles | Metrics | Comments（0）

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Unconditionally Stable Pressure-Correction Schemes for a Nonlinear Fluid-Structure Interaction Model Ying He, Jie Shen Communications on Applied Mathematics and Computation    2019, 1 (1): 61-80.   DOI: 10.1007/s42967-019-0004-0 Abstract （2850）      PDF       Knowledge map    Save We consider in this paper numerical approximation of a nonlinear fluid-structure interaction (FSI) model with a fixed interface. We construct a new class of pressure-correction schemes for the FSI problem, and prove rigorously that they are unconditionally stable. These schemes are computationally very efficient, as they lead to, at each time step, a coupled linear elliptic system for the velocity and displacement in the whole region and a discrete Poisson equation in the fluid region. Reference | Related Articles | Metrics | Comments（0）

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A Cubic H3-Nonconforming Finite Element Jun Hu, Shangyou Zhang Communications on Applied Mathematics and Computation    2019, 1 (1): 81-100.   DOI: 10.1007/s42967-019-0009-8 Abstract （941）      PDF       Knowledge map    Save The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic H3-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests. Reference | Related Articles | Metrics | Comments（0）

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Superconvergence of Energy-Conserving Discontinuous Galerkin Methods for Linear Hyperbolic Equations Yong Liu, Chi-Wang Shu, Mengping Zhang Communications on Applied Mathematics and Computation    2019, 1 (1): 101-116.   DOI: 10.1007/s42967-019-0006-y Abstract （1027）      PDF       Knowledge map    Save In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in[18] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be k + 2 when piecewise Pk polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise Pk polynomials with arbitrary k ≥ 1. Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of k + 1 and k + 2, respectively. We also prove, under suitable choice of initial discretization, a (2k + 1)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results. Reference | Related Articles | Metrics | Comments（0）

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Multiscale Radiative Transfer in Cylindrical Coordinates Wenjun Sun, Song Jiang, Kun Xu Communications on Applied Mathematics and Computation    2019, 1 (1): 117-139.   DOI: 10.1007/s42967-019-0007-x Abstract （1231）      PDF       Knowledge map    Save The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this paper, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon's mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale-dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result. Reference | Related Articles | Metrics | Comments（0）

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Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems Hui Xie, Xuejun Xu Communications on Applied Mathematics and Computation    2019, 1 (1): 141-165.   DOI: 10.1007/s42967-019-0005-z Abstract （2819）      PDF       Knowledge map    Save In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale H(div) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this H(div) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart-Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale H(div) system is bounded by C(1 + (H2/δ2))(1 + log4(H/h)), where δ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner. Reference | Related Articles | Metrics | Comments（0）

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A Two-Level Additive Schwarz Preconditioner for Local C0 Discontinuous Galerkin Methods of Kirchhof Plates Jianguo Huang, Xuehai Huang Communications on Applied Mathematics and Computation    2019, 1 (2): 167-185.   DOI: 10.1007/s42967-019-0003-1 Abstract （13485）      PDF       Knowledge map    Save A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C0 discontinuous Galerkin (LCDG) method of Kirchhof plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by O(1 + (H4/δ4)), where H is the diameter of the subdomains and δ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as O(1 + (H3/δ3)). At last, some numerical results are reported to demonstrate the high efciency of the two-level additive Schwarz preconditioner. Reference | Related Articles | Metrics | Comments（0）

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A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations Wenjie Liu, Li-Lian Wang, Shuhuang Xiang Communications on Applied Mathematics and Computation    2019, 1 (2): 207-230.   DOI: 10.1007/s42967-019-00012-1 Abstract （8805）      PDF       Knowledge map    Save In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial diferential equations (PDEs). Diferent from many other approaches, the nonstandard singular basis functions are defned from some generalised Birkhof interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems. Reference | Related Articles | Metrics | Comments（0）

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Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker-Planck Equations Tao Sun, Tian-jun Wang Communications on Applied Mathematics and Computation    2019, 1 (2): 231-252.   DOI: 10.1007/s42967-019-00013-0 Abstract （826）      PDF       Knowledge map    Save Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efciency of the scheme and conform well to theoretical analysis. Reference | Related Articles | Metrics | Comments（0）

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On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method Wen-Ting Wu Communications on Applied Mathematics and Computation    2019, 1 (2): 263-282.   DOI: 10.1007/s42967-019-00015-y Abstract （1302）      PDF       Knowledge map    Save For an upper bound of the spectral radius of the QHSS (quasi Hermitian and skew-Hermitian splitting) iteration matrix which can also bound the contraction factor of the QHSS iteration method, we give its minimum point under the conditions which guarantee that the upper bound is strictly less than one. This provides a good choice of the involved iteration parameters, so that the convergence rate of the QHSS iteration method can be signifcantly improved. Reference | Related Articles | Metrics | Comments（0）

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Jacobi-Sobolev Orthogonal Polynomials and Spectral Methods for Elliptic Boundary Value Problems Xuhong Yu, Zhongqing Wang, Huiyuan Li Communications on Applied Mathematics and Computation    2019, 1 (2): 283-308.   DOI: 10.1007/s42967-019-00016-x Abstract （3433）      PDF       Knowledge map    Save Generalized Jacobi polynomials with indexes α, β ∈ R are introduced and some basic properties are established. As examples of applications, the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems, the Jacobi-Sobolev orthogonal basis functions are constructed, which allow the exact solutions and the approximate solutions to be represented in the forms of infnite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the efectiveness and the spectral accuracy. Reference | Related Articles | Metrics | Comments（0）

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An Adaptive hp-DG-FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case Paola Antonietti, Claudio Canuto, Marco Verani Communications on Applied Mathematics and Computation    2019, 1 (3): 309-331.   DOI: 10.1007/s42967-019-00026-9 Abstract （2562）      PDF       Knowledge map    Save We propose and analyze an hp-adaptive DG-FEM algorithm, termed hp-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines:one hinges on Binev's algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the discrete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement. Reference | Related Articles | Metrics | Comments（0）

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C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations Huiyuan Li, Weikun Shan, Zhimin Zhang Communications on Applied Mathematics and Computation    2019, 1 (3): 403-434.   DOI: 10.1007/s42967-019-00041-w Abstract （15895）      PDF       Knowledge map    Save This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method. Reference | Related Articles | Metrics | Comments（0）

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A Note on the Adaptive Simpler Block GMRES Method Qiaohua Liu, Lei Yao, Aijing Liu Communications on Applied Mathematics and Computation    2019, 1 (3): 435-447.   DOI: 10.1007/s42967-019-00022-z Abstract （16575）      PDF       Knowledge map    Save The adaptive simpler block GMRES method was investigated by Zhong et al. (J Comput Appl Math 282:139-156, 2015) where the condition number of the adaptively chosen basis for the Krylov subspace was evaluated. In this paper, the new upper bound for the condition number is investigated. Numerical tests show that the new upper bound is tighter. Reference | Related Articles | Metrics | Comments（0）

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Sequential Approximation of Functions in Sobolev Spaces Using Random Samples Kailiang Wu, Dongbin Xiu Communications on Applied Mathematics and Computation    2019, 1 (3): 449-466.   DOI: 10.1007/s42967-019-00028-7 Abstract （985）      PDF       Knowledge map    Save We present an iterative algorithm for approximating an unknown function sequentially using random samples of the function values and gradients. This is an extension of the recently developed sequential approximation (SA) method, which approximates a target function using samples of function values only. The current paper extends the development of the SA methods to the Sobolev space and allows the use of gradient information naturally. The algorithm is easy to implement, as it requires only vector operations and does not involve any matrices. We present tight error bound of the algorithm, and derive an optimal sampling probability measure that results in fastest error convergence. Numerical examples are provided to verify the theoretical error analysis and the effectiveness of the proposed SA algorithm. Reference | Related Articles | Metrics | Comments（0）

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A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations Liying Zhang, Chuchu Chen, Jialin Hong, Lihai Ji Communications on Applied Mathematics and Computation    2019, 1 (3): 467-501.   DOI: 10.1007/s42967-019-00017-w Abstract （2382）      PDF       Knowledge map    Save Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al. (A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included. Reference | Related Articles | Metrics | Comments（0）

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Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models Qiang Du, Jan S. Hesthaven, Changpin Li, Chi, Wang Shu, Tao Tang Communications on Applied Mathematics and Computation    2019, 1 (4): 503-504.   DOI: 10.1007/s42967-019-00045-6 Abstract （1248）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0）

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A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions Kamran Kazmi, Abdul Khaliq Communications on Applied Mathematics and Computation    2019, 1 (4): 525-544.   DOI: 10.1007/s42967-019-00030-z Abstract （858）      PDF       Knowledge map    Save A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically. Reference | Related Articles | Metrics | Comments（0）