Asymptotic analysis of the RS-IMEX scheme for the shallow water equations in one space dimension*

In this article, we analyze a recently-presented scheme for singularly-perturbed systems of balance laws, the so-called Reference Solution Implicit Explicit scheme. RS-IMEX scheme’s bottom-line is to use the Taylor expansion of the flux function and the source term around a reference solution (typically the asymptotic limit or an equilibrium solution) to decompose the flux and the source into stiff and non-stiff parts so that the resulting IMEX scheme is Asymptotic Preserving (AP) w.r.t. the singular parameter as → 0. After a brief introduction to the scheme, we prove the asymptotic consistency, asymptotic `2-stability, solvability and well-balancing of the scheme for the case of the one-dimensional shallow water equations and with two reference solutions (the lake at rest and the zero-Froude limit). Thus, the scheme is AP and can be used for flows with various Froude numbers. Finally we will test the scheme numerically for several test cases to show the quality of the solutions and confirm the analysis.