Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

doi:10.1007/s42967-020-00116-z

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 319-352.doi: 10.1007/s42967-020-00116-z

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Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Yuan Xu, Qiang Zhang

Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu, China

收稿日期:2020-08-31 修回日期:2020-11-20 出版日期:2022-03-20 发布日期:2022-03-01
通讯作者: Qiang Zhang, Yuan Xu E-mail:qzh@nju.edu.cn;dz1721005@smail.nju.edu.cn

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Yuan Xu, Qiang Zhang

Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu, China

Received:2020-08-31 Revised:2020-11-20 Online:2022-03-20 Published:2022-03-01
Contact: Qiang Zhang, Yuan Xu E-mail:qzh@nju.edu.cn;dz1721005@smail.nju.edu.cn

摘要/Abstract

摘要： In this paper, we shall establish the superconvergence properties of the Runge-Kutta discontinuous Galerkin method for solving two-dimensional linear constant hyperbolic equation, where the upwind-biased numerical fux is used. By suitably defning the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or diferent directions, we obtain the superconvergence results on the node averages, the numerical fuxes, the cell averages, the solution and the spatial derivatives. The superconvergence properties in space are preserved as the semi-discrete method, and time discretization solely produces an optimal order error in time. Some numerical experiments also are given.

关键词: Runge-Kutta discontinuous Galerkin method, Upwind-biased fux, Superconvergence analysis, Hyperbolic equation, Two dimensions

Abstract: In this paper, we shall establish the superconvergence properties of the Runge-Kutta discontinuous Galerkin method for solving two-dimensional linear constant hyperbolic equation, where the upwind-biased numerical fux is used. By suitably defning the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or diferent directions, we obtain the superconvergence results on the node averages, the numerical fuxes, the cell averages, the solution and the spatial derivatives. The superconvergence properties in space are preserved as the semi-discrete method, and time discretization solely produces an optimal order error in time. Some numerical experiments also are given.

Key words: Runge-Kutta discontinuous Galerkin method, Upwind-biased fux, Superconvergence analysis, Hyperbolic equation, Two dimensions

中图分类号:

Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 319-352.

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Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

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