×

zbMATH — the first resource for mathematics

The Runge-Kutta discontinuous Galerkin method for conservation laws. I: Multidimensional systems. (English) Zbl 0920.65059
The paper is devoted to the construction and investigation of the so-called Runge-Kutta discontinuous Galerkin method for the numerical solution of hyperbolic conservation laws. Thereby, the method presented in previous publications is extended to systems of multidimensional nonlinear conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two-dimensional Euler equations of gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles on the quality of the approximation.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics, general
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atkins, H.L.; Shu, C.-W., Quadrature-free implementation of discontinuous Galerkin methods for hyperbolic equations, (1996)
[2] Bar-Yoseph, P., Space-time discontinuous finite element approximations for multi-dimensional nonlinear hyperbolic systems, Comput. mech., 5, 145, (1989) · Zbl 0697.65073
[3] Bar-Yoseph, P.; Elata, D., An efficientL2, Int. J. numer. methods eng., 29, 1229, (1990) · Zbl 0714.73068
[4] F. Bassi, S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys. · Zbl 0902.76056
[5] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier – stokes equations, J. comput. phys., 131, 267, (1997) · Zbl 0871.76040
[6] Bazhenova, S.; Gvozdeva, L.; Nettleton, M., Unsteady interactions of shock waves, Prog. aerosp. sci., 21, 249, (1984)
[7] Berger, M.; Colella, A., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[8] Bey, K.S.; Oden, J.T., A runge – kutta discontinuous Galerkin finite element method for high speed flows, AIAA 10th computational fluid dynamics conference, (1991)
[9] Biswas, R.; Devine, K.D.; Flaherty, J., Parallel, adaptive finite element methods for conservation laws, Applied numerical mathematics, 14, 255, (1994) · Zbl 0826.65084
[10] Carey, G.F.; Oden, J.T., Finite elements: computational aspects, III, (1984) · Zbl 0558.73064
[11] G. Chavent, B. Cockburn, The local projectionP0P1, M, 2, AN, 23, 565, 1989 · Zbl 0715.65079
[12] Ciarlet, P., The finite element method for elliptic problems, (1975)
[13] B. Cockburn, C. W. Shu, The Runge-Kutta local projectionP1, M, 2, AN, 25, 337, 1991 · Zbl 0732.65094
[14] Cockburn, B.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework, Math. comp., 52, 411, (1989) · Zbl 0662.65083
[15] 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, (1989) · Zbl 0677.65093
[16] Cockburn, B.; Hou, S.; Shu, C.W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comp., 54, 545, (1990) · Zbl 0695.65066
[17] Cockburn, B.; Shu, C.W., TheP1, (1991)
[18] B. Cockburn, C. W. Shu, The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM J. Numer. Anal. · Zbl 0927.65118
[19] deCougny, H.L.; Devine, K.D.; Flaherty, J.E.; Loy, R.M.; Özturan, C.; Shephard, M.S., High-order accurate discontinuous discontinuous finite element solution of the 2D Euler equations, Applied numerical mathematics, 16, 157, (1994) · Zbl 0818.65099
[20] Devine, K.D.; Flaherty, J.E.; Loy, R.M.; Wheat, S.R., Parallel partitioning strategies for the adaptive solution of conservation laws, (1994)
[21] Devine, K.D.; Flaherty, J.E.; Wheat, S.R.; Maccabe, A.B., A massively parallel adaptive finite element method with dynamic load balancing, (1993)
[22] Hillier, R., Computation of shock wave diffraction at a ninety degrees convex edge, Shock waves, 1, 89, (1991) · Zbl 0825.76402
[23] Hou, S., A finite element method for conservation laws: multidimensional case, (1991)
[24] Jiang, G.; Shu, C.-W., On cell entropy inequality for discontinuous Galerkin methods, Math. comp., 62, 531, (1994) · Zbl 0801.65098
[25] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. comp., 46, 1, (1986) · Zbl 0618.65105
[26] LeSaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport equation, 89, (1974)
[27] R. B. Lowrie, P. L. Roe, B. van Leer, Space-time discontinuous Galerkin: I. Theory and properties · Zbl 0947.76055
[28] Özturan, C.; deCougny, H.L.; Shephard, M.S.; Flaherty, J.E., Parallel adaptive mesh refinement and redistribution on distributed memory computers, Comput. methods appl. mech. engrg., 119, 123, (1994) · Zbl 0851.73068
[29] Peterson, T., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. numer. anal., 28, 133, (1991) · Zbl 0729.65085
[30] Quirk, J., A construction to the great Riemann solver debate, Int. J. numer. meth. fluids, 18, 555, (1994) · Zbl 0794.76061
[31] Reed, W.H.; Hill, T.R., Triangular mesh methods for the neutron transport equation, (1973)
[32] Richter, G.R., An optimal-order error estimate for the discontinuous Galerkin method, Math. comp., 50, 75, (1988) · Zbl 0643.65059
[33] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. comp., 49, 105, (1987) · Zbl 0628.65075
[34] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. stat. comput., 9, 1073, (1988) · Zbl 0662.65081
[35] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[36] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[37] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115, (1984) · Zbl 0573.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.