Structure-preserving shock-capturing methods: late-time asymptotics, curved geometry, small-scale dissipation, and nonconservative products
Résumé
We consider weak solutions to nonlinear hyperbolic systems of conservation laws arising in compressible fluid dynamics and we describe recent work on the design of structure-preserving numerical methods. We focus on preserving, on one hand, the late-time asymptotics of solutions and, on the other hand, the geometrical effects that arise in certain applications involving curved space. First, we study here nonlinear hyperbolic systems with stiff relaxation in the late time regime. By performing a singular analysis based on a Chapman– Enskog expansion, we derive an effective system of parabolic type and we introduce a broad class of finite volume schemes which are consistent and accurate even for asymptotically late times. Second, for nonlinear hyperbolic conservation laws posed on a curved manifold, we formulate geometrically consistent finite volume schemes and, by generalizing the Cockburn–Coquel–LeFloch's theorem, we establish the strong convergence of the approximate solutions toward entropy solutions.
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