High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

TL;DR

This paper introduces high-order semi-implicit WENO schemes for the shallow water equations that are well-balanced, asymptotic preserving, and accurate across all Froude numbers, including the zero limit.

Contribution

The paper develops a novel high-order semi-implicit WENO scheme that is well-balanced, asymptotic preserving, and effective for all Froude numbers in shallow water equations.

Findings

The schemes achieve high-order accuracy in numerical tests.

They effectively preserve steady states and capture small perturbations.

The methods demonstrate robustness across a range of Froude numbers.

Abstract

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.