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A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes. (English) Zbl 1083.65093
Summary: 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, total variation diminishing (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.
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 second-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, nonoscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
HLLE; HE-E1GODF
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[1] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comp., 54, 545-581, (1990) · Zbl 0695.65066
[2] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. comput. phys., 84, 90-113, (1989) · Zbl 0677.65093
[3] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. comp., 52, 411-435, (1989) · Zbl 0662.65083
[4] Cockburn, B.; Shu, C.-W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. model. numer. anal. (M^{2}AN),, 25, 337-361, (1991) · Zbl 0732.65094
[5] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. comput. phys., 141, 199-224, (1998) · Zbl 0920.65059
[6] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin method for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[7] Cockburn, B.; Shu, C.-W., Foreword (for the special issue on discontinuous Galerkin methods), J. sci. comput., 22-23, 1-2, (2005)
[8] Engquist, B.; Osher, S., One sided difference approximation for nonlinear conservation laws, Math. comp., 36, 321-351, (1981) · Zbl 0469.65067
[9] Godunov, S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. sbornik, 47, 271-306, (1959) · Zbl 0171.46204
[10] Harten, A.; Engquist, B.; Osher, S.; Chakravathy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067
[11] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-61, (1983) · Zbl 0565.65051
[12] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[13] Krivodonova, L.; Xin, J.; Remacle, J.-F.; Chevaugeon, N.; Flaherty, J.E., Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. numer. math., 48, 323-338, (2004) · Zbl 1038.65096
[14] Osher, S.; Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. comp., 38, 339-374, (1982) · Zbl 0483.65055
[15] Qiu, J.; Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. sci. comput., 26, 907-929, (2005) · Zbl 1077.65109
[16] J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using WENO limiters, SIAM J. Sci. Comput., in press. · Zbl 1092.65084
[17] W.H. Reed, T.R. Hill, Triangular mesh methods for neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
[18] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. comp., 49, 105-121, (1987) · Zbl 0628.65075
[19] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[20] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[21] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. comput. phys., 83, 32-78, (1989) · Zbl 0674.65061
[22] Titarev, V.A.; Toro, E.F., Finite-volume WENO schemes for three-dimensional conservation laws, J. comput. phys., 201, 238-260, (2004) · Zbl 1059.65078
[23] Titarev, V.A.; Toro, E.F., WENO schemes based on upwind and centred TVD fluxes, Comput. fluids, 34, 705-720, (2005) · Zbl 1134.65361
[24] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, a practical introduction, (1997), Springer Berlin · Zbl 0888.76001
[25] E.F. Toro, Multi-stage predictor-corrector fluxes for hyperbolic equations, Preprint NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK.
[26] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the harten-Lax-Van leer Riemann solver, J. shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[27] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
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