On a Cell Entropy Inequality for Discontinuous Galerkin Methods

Abstract

Abstract. We prove a cell entropy inequality for a class of high-order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one-dimensional scalar convex case.

L2stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods

“…However, we cannot prove a similar L 2 stability result for the central DG scheme when applied to the nonlinear scalar conservation law (1.1), even though the proof of Theorem 2.1 can be easily generalized to multi-dimensional central DG schemes for linear equations. This is in contrary to the cell entropy inequality for regular DG schemes, which holds for arbitrary nonlinear scalar conservation laws [6].…”

“…The proof of Theorem 2.1 is similar to the proof of the cell entropy inequality for the regular DG method in [6]. However, we cannot prove a similar L 2 stability result for the central DG scheme when applied to the nonlinear scalar conservation law (1.1), even though the proof of Theorem 2.1 can be easily generalized to multi-dimensional central DG schemes for linear equations.…”

“…The organization of the paper is as follows. In Section 2, we first analyze the L 2 stability and give an a priori error estimate for the semi-discrete central DG scheme (1.5)-(1.6) for the linear hyperbolic equation, using similar techniques of stability and error analysis for standard DG methods [3,6]. We then perform a Fourier analysis for the semi-discrete central DG scheme (1.5)-(1.6) for the linear hyperbolic equation with uniform meshes for piecewise constant and linear elements, using the techniques in [13].…”

“…We study the L 2 stability of the central DG scheme (1.5)-(1.6) for the equation (2.1) in Section 2.1, and compare the result with that for the regular DG scheme in [6]. In Section 2.2 we provide an L 2 a priori error estimate for smooth solutions, and compare the result with that for the regular DG scheme in [3].…”

A MUSCL method satisfying all the numerical entropy inequalities

“…J. Jaffre, C. Johnson and A. Szepessy [12] have developed a high-order multidimensional discontinuous Galerkin method, which satisfies all the entropy conditions, but again with a nonhomogeneous artificial viscosity term. In a simpler context, G. Jiang and C.-W. Shu [13] have presented a simple approach to get this inequality without unnatural limitation or viscosity.…”