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A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods

Published online by Cambridge University Press:  03 June 2015

Hongqiang Zhu ,
Yue Cheng  and
Jianxian Qiu
Hongqiang Zhu*
Affiliation:
School of Natural Science, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210023, China
Yue Cheng*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China Baidu, Inc. Baidu Campus, No. 10, Shangdi 10th Street, Haidian District, Beijing 100085, China
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
*
URL: http://ccam.xmu.edu.cn/teacher/jxqiu, Email: zhuhq@njupt.edu.cn
Email: chengyue127@gmail.com
Corresponding author. Email: jxqiu@xmu.edu.cn
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Abstract

Discontinuities usually appear in solutions of nonlinear conservation laws even though the initial condition is smooth, which leads to great difficulty in computing these solutions numerically. The Runge-Kutta discontinuous Galerkin (RKDG) methods are efficient methods for solving nonlinear conservation laws, which are high-order accurate and highly parallelizable, and can be easily used to handle complicated geometries and boundary conditions. An important component of RKDG methods for solving nonlinear conservation laws with strong discontinuities in the solution is a nonlinear limiter, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods. A limiter contains two parts, first to identify the “troubled cells”, namely, those cells which might need the limiting procedure, then to replace the solution polynomials in those troubled cells by reconstructed polynomials which maintain the original cell averages (conservation). [SIAM J. Sci. Comput., 26 (2005), pp. 995–1013.] focused on discussing the first part of limiters. In this paper, focused on the second part, we will systematically investigate and compare a few different reconstruction strategies with an objective of obtaining the most efficient and reliable reconstruction strategy. This work can help with the choosing of right limiters so one can resolve sharper discontinuities, get better numerical solutions and save the computational cost.

Keywords

65M6065M9935L65Limiterdiscontinuous Galerkin methodhyperbolic conservation laws
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 3 , June 2013 , pp. 365 - 390
Copyright
Copyright © Global-Science Press 2013

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