Analysis and entropy stability of the line-based discontinuous Galerkin method

Abstract

We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions.

25 pages, 7 figures

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• On the robustness and performance of entropy stable discontinuous collocation methods for the compressible Navier-Stokes equations
• Formulation of Entropy-Stable schemes for the multicomponent compressible Euler equations
• Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems
• Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra
• Entropy stabilization and property-preserving limiters for discontinuous Galerkin discretizations of nonlinear hyperbolic equations
• Entropy-Stable Schemes in the Low-Mach-Number Regime: Flux-Preconditioning, Entropy Breakdowns, and Entropy Transfers