Table of Content

20 December 2020, Volume 2 Issue 4

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ORIGINAL PAPER

Single-Step Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for 1-D Euler Equations

Jayesh Badwaik, Praveen Chandrashekar, Christian Klingenberg

2020, 2(4):  541-579.  doi:10.1007/s42967-019-00054-5

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We propose an explicit, single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations. The grid is moved with the local fuid velocity modifed by some smoothing, which is found to considerably reduce the numerical dissipation introduced by Riemann solvers. The scheme preserves constant states for any mesh motion and we also study its positivity preservation property. Local grid refnement and coarsening are performed to maintain the mesh quality and avoid the appearance of very small or large cells. Second, higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.

Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Diference Summation by Parts Operators

Hendrik Ranocha, Katharina Ostaszewski, Philip Heinisch

2020, 2(4):  581-611.  doi:10.1007/s42967-019-00057-2

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In this article, discrete variants of several results from vector calculus are studied for classical fnite diference summation by parts operators in two and three space dimensions. It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector felds cannot hold discretely because of grid oscillations, which are characterised explicitly. This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition. Nevertheless, iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are proposed and applied successfully. In numerical experiments, the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other frst-order partial diferential equations. Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics, applications to the discrete analysis of magnetohydrodynamic (MHD) wave modes are presented and discussed.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Difusion Equation

Min Zhang, Yang Liu, Hong Li

2020, 2(4):  613-640.  doi:10.1007/s42967-019-00058-1

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In this article, some high-order local discontinuous Galerkin (LDG) schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimensional nonlinear fractional difusion equation. The unconditional stability of the LDG scheme is proved, and an a priori error estimate with O(h^k+1 + △t²) is derived, where k ≥ 0 denotes the index of the basis function. Extensive numerical results with Q^k(k=0, 1, 2, 3) elements are provided to confrm our theoretical results, which also show that the secondorder convergence rate in time is not impacted by the changed parameter θ.

Some Criteria for H-Tensors

Guangbin Wang, Fuping Tan

2020, 2(4):  641-651.  doi:10.1007/s42967-019-00059-0

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The H-tensor is a new developed concept in tensor analysis and it is an extension of the M -tensor. In this paper, we present some criteria for identifying nonsingular H-tensors and give two numerical examples.

Second-Order Finite Diference/Spectral Element Formulation for Solving the Fractional Advection-Difusion Equation

Mostafa Abbaszadeh, Hanieh Amjadian

2020, 2(4):  653-669.  doi:10.1007/s42967-020-00060-y

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The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-difusion equation. The time variable has been discretized by a second-order fnite diference procedure. The stability and the convergence of the semi-discrete formula have been proven. Then, the spatial variable of the main PDEs is approximated by the spectral element method. The convergence order of the fully discrete scheme is studied. The basis functions of the spectral element method are based upon a class of Legendre polynomials. The numerical experiments confrm the theoretical results.

A Finite Diference Method for Space Fractional Diferential Equations with Variable Difusivity Coefcient

K. A. Mustapha, K. M. Furati, O. M. Knio, O. P. Le Maître

2020, 2(4):  671-688.  doi:10.1007/s42967-020-00066-6

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Anomalous difusion is a phenomenon that cannot be modeled accurately by second-order difusion equations, but is better described by fractional difusion models. The nonlocal nature of the fractional difusion operators makes substantially more difcult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the frst fnite diference method for solving variable-coefcient onedimensional (steady state) fractional diferential equations (DEs) with two-sided fractional derivatives (FDs). The proposed scheme combines frst-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the frst-order backward Euler method. Our scheme reduces to the standard second-order central diference in the absence of FDs. The existence and uniqueness of the numerical solution are proved, and truncation errors of order h are demonstrated (h denotes the maximum space step size). The numerical tests illustrate the global O(h) accuracy, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Difusion Equations

Somayeh Yeganeh, Reza Mokhtari, Jan S. Hesthaven

2020, 2(4):  689-709.  doi:10.1007/s42967-020-00065-7

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For two-dimensional (2D) time fractional difusion equations, we construct a numerical method based on a local discontinuous Galerkin (LDG) method in space and a fnite diference scheme in time. We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable. Numerical results indicate the efectiveness and accuracy of the method and confrm the analysis.