High-order fully well-balanced Lagrange-projection scheme for shallow water
Résumé
In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [M.J. Castro et al., in Handbook of Numerical Analysis, 18:131175, 2017], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.
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