A posteriori Finite-Volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Computational Physics Année : 2022

A posteriori Finite-Volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations

Ali Haidar
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Fabien Marche
François Vilar

Résumé

We design and analyze a new discretization method for the nonlinear shallow water equations, which is based on an equivalent representation of arbitrary high-order Discontinuous Galerkin (DG) schemes through piecewise constant modes on a sub-grid, together with a selective a posteriori local correction of the sub-interface reconstructed flux. This new approach, based on [F. Vilar, J. Comput. Phys., 387:245-279, 2019], allows to combine at the subcell scale the excellent robustness properties of the Finite-Volume (FV) lowest-order method and the high-order accuracy of the DG method. For any order of polynomial approximation, the resulting algorithm is shown to: (i) accurately handle strong shocks with no robustness issues; (ii) ensure the preservation of the water height positivity at the subcell level; (iii) preserve the class of motionless steady states (well-balancing); (iv) retain the highly accurate subcell resolution of DG schemes. These assets are numerically illustrated through an extensive set of test-cases, with a particular emphasize put on the use of very-high order polynomial approximations on coarse grids.
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Dates et versions

hal-03549725 , version 1 (31-01-2022)

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Ali Haidar, Fabien Marche, François Vilar. A posteriori Finite-Volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations. Journal of Computational Physics, 2022, 452, pp.110902. ⟨10.1016/j.jcp.2021.110902⟩. ⟨hal-03549725⟩
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