An entropy-stable, positivity-preserving Godunov-type scheme for multidimensional hyperbolic systems of conservation laws on unstructured grids was presented by Gallice et al. in [1]. A specific feature of their Riemann solver is coupling all cells in the vicinity of the current one thanks to a nodal parameter: the velocity of the nodes. Consequently, this Riemann solver is no longer 1D across one edge. Instead, it encounters genuine multidimensional effects. We extended this work [1] to handle source terms, with a specific application to the shallow water system. The scheme we obtain is well-balanced in 1D and 2D. The numerical scheme appears to be insensitive to the numerical instability known as Carbuncle in supersonic and hypersonic flows. As a further research path, we investigate whether the knowledge of multidimensional effects can improve numerical results for low-Mach flows. A new version of the method, based on a nodal pressure, is currently being explored and seems to yield satisfactory results.
References [1] G. Gallice, A. Chan, R. Loubère, P.-H. Maire. Entropy Stable and Positivity Preserving Godunov-Type Schemes for Multidimensional Hyperbolic Systems on Unstructured Grid. Journal of Computational Physics, Volume 468, 2022, 111493, ISSN 0021-9991, https://doi.org/10.1016/j.jcp.2022.111493.