A Semi-Implicit Finite Volume Relaxation Scheme for 2D Shallow-Water System
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This work presents a two-dimensional extension of the semi-implicit finite-volume relaxation scheme originally introduced for the one-dimensional shallow-water equations. The goal is to retain the main advantages of the 1D formulation---in particular, outperform in low-Froude regimes and well-balanced properties---while providing an efficient solver for 2D flows. First, the core ideas behind the relaxation approach are revisited. The scheme consists of a splitting that separates the acoustic and transport subsystems. The acoustic step, built from the Riemann invariants of the associated relaxed system, targets the gravity waves through an implicit update, while the transport step advances the material waves explicitly. This decomposition provides a clear route to construct the numerical fluxes and leads to a two-dimensional scheme. Finally, a consistent 2D extension is designed within a dimensional-splitting framework by coupling the relevant relaxation variables across coordinate directions. The resulting solver provides an efficient 2D method that preserves the semi-implicit character and the well-balanced features of the original approach.
