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Mortar-based entropy-stable discontinuous Galerkin methods on non-conforming quadrilateral and hexahedral meshes. (English) Zbl 1502.65115

Summary: High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points [J. Chan et al., SIAM J. Sci. Comput. 41, No. 5, A2938–A2966 (2019; Zbl 1435.65172)]. In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between “lines” of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments for the compressible Euler equations in two and three dimensions confirm the stability and accuracy of this approach.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N08 Finite volume methods for boundary value problems involving PDEs
65D30 Numerical integration
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q31 Euler equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
76M20 Finite difference methods applied to problems in fluid mechanics

Citations:

Zbl 1435.65172
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References:

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