Asymptotic Preserving Finite Volume Schemes for the Singularly-perturbed Shallow Water Equations with Source Terms

Abstract

The shallow water equations are of quite an importance for modelling oceanic flows as a simple approximation of the water wave equations, which describe the gravity-driven free surface flows, when the fluid is incompressible, homogeneous, inviscid, and the pressure is only hydrostatic, owing to the shallowness assumption, that is the horizontal length scale is much larger than the vertical one. For the shallow water equations, the Froude number characterises the dominance of advective modes compared to gravity (acoustic) modes as the ratio of the bulk velocity to the speed of gravity waves. For the large-scale oceanic phenomena, the Froude number is often small; so, the gravity waves are too fast to contribute to the bulk motion, i.e., they do not affect the solution of the large-scale macroscopic model. For a time-explicit numerical treatment, though, one should devise a method to tackle these fast waves to avoid high computational costs as they restrict the time step through the Courant--Friedrichs--Lewy (CFL) condition. The approach considered throughout this manuscript is to decompose the system into slow and fast parts and to employ an implicit-explicit (IMEX) strategy, i.e., to treat the fast part implicitly and the slow part explicitly. In addition to the efficiency problem attached to this singularly-perturbed system, one should be careful about the limiting scheme, i.e., if the scheme provides a consistent and stable approximation of the zero-Froude system (lake equations). Even if the convergence to the limit can be shown for the continuous model, preserving such a convergence for the discrete (numerical) model, along with stability and consistency, is by no means trivial and should be carefully analysed. This motivates adopting the framework of asymptotic preserving (AP) schemes introduced by [Jin, SIAM J. Sci. Comp. 21(2) (1999), pp. 441--454], with the Froude number as the scaling singular parameter. AP schemes are defined as schemes mimicking such a convergence to the limit for the discrete model, e.g., in virtue of uniform consistency and stability. In this manuscript, we consider two IMEX flux-splitting finite volume schemes for the shallow water equations with uniform consistency and stability w.r.t. the Froude number: the Lagrange-projection IMEX scheme and the reference solution IMEX scheme. The LP-IMEX scheme is a Godunov-type scheme, which decomposes the system into the acoustic and the transport systems, and employs a Lagrangian formulation for the former. Unfortunately, it is involved in some inherent accuracy issues especially in multiple dimensions, which need to be taken care of; so, we investigate it only for the one-dimensional system. The main focus would be on the RS-IMEX scheme, which decomposes the solution into the (asymptotic) reference solution and a perturbation around it in order to split the system. We study the RS-IMEX scheme in one and two space dimensions with the bottom topography, and finally, with the additional Coriolis force. For both of these schemes, we present a (rigorous) asymptotic analysis to justify the uniform consistency and stability of the scheme w.r.t. the Froude number, and to corroborate the AP property. We also test the quality of the solutions computed by the RS-IMEX scheme in several numerical examples, particularly for the low-Froude regime.