Asymptotic preserving scheme for the shallow water equations with non-flat bottom topography and Manning friction term

TL;DR

This paper develops a high-order asymptotic preserving scheme for shallow water equations with bottom topography and Manning friction, removing penalization for improved efficiency while maintaining stability and accuracy.

Contribution

It introduces a semi-implicit IMEX-RK scheme combined with WENO reconstruction that preserves asymptotic properties without penalization, enhancing computational efficiency.

Findings

The scheme retains AP, AA, and well-balanced properties.

Implicit treatment of Manning friction is crucial for convergence.

The new method is more efficient in intermediate regimes.

Abstract

In our previous work [29], we proposed a class of high-order asymptotic preserving (AP) finite difference weighted essentially non-oscillatory (WENO) schemes for solving the shallow water equations (SWEs) with bottom topography and Manning friction, utilizing a penalization technique inspired by [6]. Although the added weighted diffusive term enhanced stability, it increased computational cost and slowed down the convergence rate in the intermediate regime between convection and diffusion. In this paper, we extend our previous study by removing the penalization while preserving the AP property. To achieve this, we employ a high order semi-implicit implicit-explicit Runge-Kutta (SI-IMEX-RK) time discretization, coupled with the high-order WENO reconstruction for first-order derivatives and a central difference scheme for second-order spatial derivatives. This combination yields a class of fully high-order schemes. Theoretical analysis and numerical experiments demonstrate that the proposed schemes retain AP, asymptotically accurate (AA) and well-balanced properties, while offering higher computational efficiency compared to our previous schemes in [29], especially in the intermediate regime between convection and diffusion. Moreover, treating the momentum in the friction terms implicitly is essential for preserving the AP property; otherwise, the scheme fails to converge to the limiting equations. This indicates that implicit treatment of Manning friction is necessary for the stability of the method.