An all-regime and well-balanced Lagrange-projection type scheme for the shallow water equations on unstructured meshes
This work provides a robust numerical method for simulating shallow water flows across all regimes, addressing the challenging low-Froude regime, which is important for environmental and engineering applications.
The authors developed a well-balanced, all-regime, and positivity-preserving numerical scheme for the shallow water equations on unstructured meshes, using a Lagrange-projection decomposition to handle low-Froude regimes effectively.
In this work, we focus on the numerical approximation of the shallow water equations in two space dimensions. Our aim is to propose a well-balanced, all-regime and positive scheme. By well-balanced, it is meant that the scheme is able to preserve the so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that the scheme is able to deal with all flow regimes, including the low-Froude regime which is known to be challenging when using usual Godunov-type finite volume schemes. At last, the scheme should be positive which means that the water height stays positive for all time. Our approach is based on a Lagrange-projection decomposition which allows to naturally decouple the acoustic and transport terms. Numerical experiments on unstructured meshes illustrate the good behaviour of the scheme.