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Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. (English) Zbl 1380.65253
Summary: It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws [G. Jiang and the second author, Math. Comput. 62, No. 206, 531–538 (1994; Zbl 0801.65098)] and symmetric hyperbolic systems [S. Hou and X.-D. Liu, J. Sci. Comput. 31, No. 1–2, 127–151 (2007; Zbl 1152.76433)], in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and/or when the integration is approximated by a numerical quadrature. In this paper, we develop a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in [M. H. Carpenter et al., SIAM J. Sci. Comput. 36, No. 5, B835–B867 (2014; Zbl 1457.65140); G. J. Gassner, SIAM J. Sci. Comput. 35, No. 3, A1233–A1253 (2013; Zbl 1275.65065)]. The main ingredients are summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy conservative flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Extensive numerical experiments are performed to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35D30 Weak solutions to PDEs
35R45 Partial differential inequalities and systems of partial differential inequalities
Software:
HLLE; HE-E1GODF; Gmsh
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References:
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