In this paper, we study formally high-order accurate discontinuous Galerkin methods on general arbitrary grid for multi-dimensional hyperbolic systems of conservation laws [Cockburn, B., and Shu, C.-W. (1989, Math. Comput. 52, 411–435, 1998, J. Comput. Phys. 141, 199–224); Cockburn et al. (1989, J. Comput. Phys. 84, 90–113; 1990, Math. Comput. 54, 545–581). We extend the notion of E-flux [Osher (1985) SIAM J. Numer. Anal. 22, 947–961] from scalar to system, and found that after flux splitting upwind flux [Cockburn et al. (1989) J. Comput. Phys. 84, 90–113] is a Riemann solver free E-flux for systems. Therefore, we are able to show that the discontinuous Galerkin methods satisfy a cell entropy inequality for square entropy (in semidiscrete sense) if the multi-dimensional systems are symmetric. Similar result [Jiang and Shu (1994) Math. Comput. 62, 531–538] was obtained for scalar equations in multi-dimensions. We also developed a second-order finite difference version of the discontinuous Galerkin methods. Numerical experiments have been obtained with excellent results.
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References
Cockburn B., Shu C.-W. (1989). TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52:411–435
Cockburn B., Hou S., Shu C.-W. (1990). The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54:545–581
Cockburn B., Lin S.-Y., Shu C.-W. (1989). TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84:90–113
Cockburn B., and Shu C.-W. (1998). The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141:199–224
Jiang G.-S., Shu C.-W. (1994). On cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62:531–538
Osher S. (1985). Convergence of generalized MUSCL schemes. SIAM J. Numer. Anal. 22:947–961
Shu C.-W., and Osher S. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83:32–78
Roe P.L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43:357–372
Barth, T., and Charrier, P. (2001). Energy Stable Flux Formulas for the Discontinuous Galerkin Discretization of First Order Nonlinear Conservation Laws. NAS technical reports.
Sweby P.K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5):995–1011
Zhang T., Zheng Y. (1990). Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems. SIAM J. Math. Anal. 21(3):593–630
Schulz-Rinne C.W. (1993). Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24:76–88
Schulz-Rinne C.W., Collins J.P., Glaz H.M. (1993). Numerical solution of the Riemann problem for two-dimensional gas dynamics. SIAM J. Sci. Comput. 14(6):1394–1414
Lax P., and Liu X.-D. (1998). Solution of two dimensional Riemann problem of gas dynamics. by positive schemes. SIAM J. Sci. Compu. 19(2):319–340
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Hou, S., Liu, XD. Solutions of Multi-dimensional Hyperbolic Systems of Conservation Laws by Square Entropy Condition Satisfying Discontinuous Galerkin Method. J Sci Comput 31, 127–151 (2007). https://doi.org/10.1007/s10915-006-9105-9
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DOI: https://doi.org/10.1007/s10915-006-9105-9
Keywords
- Multi-dimensional Hyperbolic systems of conservation laws
- Discontinuous Galerkin method
- Square entropy condition
- E-flux
- Finite difference