A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes

Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVDRunge–Kutta time discretizations, and limiters. Inmost of theRKDGpapers in the literature, theLax–Friedrichs numerical flux is useddue to its simplicity, although there aremanyother numerical fluxes which couldalsobeused. In thispaper,we systematically investigate the performanceof theRKDGmethodbasedondifferent numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and second-order TVDfluxes,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, nonoscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems. 2005 Elsevier Inc. All rights reserved. AMS: 65M60; 65M99; 35L65 0021-9991/$ see front matter 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: [email protected] (J. Qiu), [email protected] (B.C. Khoo), [email protected] (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 AROGrant W911NF-04-1-0291, NSF Grant DMS-0207451 and AFOSRGrant FA9550-05-1-0123. J. Qiu et al. / Journal of Computational Physics 212 (2006) 540–565 541