A numerical study for the performance of the Runge–Kutta discontinuous Galerkin method based on different numerical fluxes Jianxian Qiu a,b,1 , Boo Cheong Khoo b,2 , Chi-Wang Shu c, * ,3 a Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, PR China b Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore c Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA Received 11 May 2005; accepted 13 July 2005 Available online 29 August 2005 Abstract Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic con- servation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Rie- mann solvers serving as numerical fluxes, TVD Runge–Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and sec- ond-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non- oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems. Ó 2005 Elsevier Inc. All rights reserved. AMS: 65M60; 65M99; 35L65 0021-9991/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2005.07.011 * Corresponding author. Tel.: +1 401 863 2549; fax: +1 401 863 1355. E-mail addresses: jxqiu@nju.edu.cn (J. Qiu), mpekbc@nus.edu.sg (B.C. Khoo), shu@dam.brown.edu (C.-W. Shu). 1 Research partially supported by NNSFC Grant 10371118, Nanjing University Talent Development Foundation and NUS Research Project R-265-000-118-112. 2 Research partially supported by NUS Research Project R-265-000-118-112. 3 Research partially supported by the Chinese Academy of Sciences while the author was in residence at the University of Science and Technology of China (Grant 2004-1-8) and at the Institute of Computational Mathematics and Scientific/Engineering Computing. Additional support is provided by ARO Grant W911NF-04-1-0291, NSF Grant DMS-0207451 and AFOSR Grant FA9550-05-1-0123. Journal of Computational Physics 212 (2006) 540–565 www.elsevier.com/locate/jcp