A Semi-Implicit Fully Exactly Well-Balanced Relaxation Scheme for the Shallow Water System

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2503-A2527, August 2024.
Abstract. This article focuses on the design of semi-implicit schemes that are fully well-balanced for the one-dimensional shallow water equations, that is, schemes that preserve all smooth steady states of the system and not just water-at-rest equilibria. The proposed methods outperform standard explicit schemes in the low-Froude regime, where the celerity is much larger than the fluid velocity, eliminating the need for a large number of iterations on large time intervals. In this work, splitting and relaxation techniques are combined in order to obtain fully well-balanced semi-implicit first and second order schemes. In contrast to recent Lagrangian-based approaches, this one allows the preservation of all the steady states while avoiding the complexities associated with Lagrangian formalism.