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.
Multi-dimensional Hyperbolic systems of conservation laws Discontinuous Galerkin method Square entropy condition E-flux Finite difference