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

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

Published online by Cambridge University Press:  07 February 2017

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:
Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, USA
Chi-Wang Shu*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, 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@sci.utah.edu (X. Zhong), shu@dam.brown.edu (C.-W. Shu), jxqiu@xmu.edu.cn (J. Qiu)
*Corresponding author. Email addresses:zhujun@nuaa.edu.cn (J. Zhu), zhongxh@sci.utah.edu (X. Zhong), shu@dam.brown.edu (C.-W. Shu), jxqiu@xmu.edu.cn (J. Qiu)
*Corresponding author. Email addresses:zhujun@nuaa.edu.cn (J. Zhu), zhongxh@sci.utah.edu (X. Zhong), shu@dam.brown.edu (C.-W. Shu), jxqiu@xmu.edu.cn (J. Qiu)
*Corresponding author. Email addresses:zhujun@nuaa.edu.cn (J. Zhu), zhongxh@sci.utah.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 generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed in [38] for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion [11] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.

Keywords

Runge-Kutta discontinuous Galerkin methodHWENO limiterunstructured mesh

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 21 , Issue 3 , March 2017 , pp. 623 - 649
Copyright
Copyright © Global-Science Press 2017 

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