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

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. • Develop a high order well balanced AP scheme for the SWEs with source terms. • Combine the WENO spatial discretization with an IMEX temporal discretization. • Achieve well-balanced properties for a still-water steady state. • Prove the asymptotic preserving and asymptotically accurate properties. • Provide ample numerical results in high and low Froude regimes with great performance.