Semi-implicit fully exactly well-balanced schemes for the two-layer shallow water system

Abstract

This work addresses the design of semi-implicit numerical schemes that are fully exactly well-balanced for the two-layer shallow water system, meaning that they are capable of preserving every possible steady state, and not only the lake-at-rest ones. The proposed approach exhibits better performance compared to standard explicit methods in low-Froude number regimes, where wave propagation speeds significantly exceed flow velocities, thereby reducing the computational cost associated with long-time simulations. The methodology relies on a combination of splitting strategies and relaxation techniques to construct first- and second-order semi-implicit schemes that satisfy the fully exactly well-balanced property.