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Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 April 2016

Jun Zhu ,
Xinghui Zhong ,
Chi-Wang Shu  and
Jianxian Qiu
Jun Zhu
Affiliation:
College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, P.R. China
Xinghui Zhong
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Chi-Wang Shu
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI02912, USA
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen, Fujian 361005, P.R. China
*
*Corresponding author.Email addresses:zhujun@nuaa.edu.cn (J. Zhu), zhongxh@math.msu.edu (X. Zhong), shu@dam.brown.edu (C.-W. Shu), jxqiu@xmu.edu.cn (J. Qiu)
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Abstract

In this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.

Keywords

Runge-Kutta discontinuous Galerkin methodHWENO limiterconservation law

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35L65: Conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 944 - 969
Copyright
Copyright © Global-Science Press 2016 

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