Adjoint-Based Adaptive Model and Discretization for Hyperbolic Systems with Relaxation

Abstract : In this work, we use an adjoint-weighted residuals method for the derivation of an a posteriori model and discretization error estimators in the approximation of solutions to hyperbolic systems with stiff relaxation source terms and multiscale relaxation rates. These systems are parts of a hierarchy of models where the solution reaches different equilibrium states associated to different relaxation mechanisms. The discretization is based on a discontinuous Galerkin method which allows to account for the local regularity of the solution during the discretization adaptation. The error estimators are then used to design an adaptive model and discretization procedure which selects locally the model, the mesh, and the order of the approximation and balances both error components. Coupling conditions at interfaces between different models are imposed through local Riemann problems to ensure the transfer of information. The reliability of the present hpm-adaptation procedure is assessed on different test cases involving a Jin--Xin relaxation system with multiscale relaxation rates, and results are compared with standard hp-adaptation.
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Dylan Dronnier, Florent Renac. Adjoint-Based Adaptive Model and Discretization for Hyperbolic Systems with Relaxation. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2019, 17 (2), pp.750-772. ⟨10.1137/18M120676X⟩. ⟨hal-02179030⟩

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