A well-balanced finite difference WENO scheme for shallow water flow model

TL;DR

This paper introduces a novel high-order finite difference WENO scheme for shallow water flow over non-flat bottoms, based on an alternative formulation that simplifies achieving well-balanced and non-oscillatory properties while preserving conservation and accuracy.

Contribution

It proposes a new WENO scheme using pre-balanced shallow water equations, enhancing stability and accuracy over existing methods.

Findings

The scheme preserves exact conservation properties.

It maintains high-order accuracy for smooth solutions.

It effectively handles discontinuities without oscillations.

Abstract

In this paper, we are concerned with the shallow water flow model over non-flat bottom topography by high-order schemes. Most of the numerical schemes in the literature are developed from the original mathematical model of the shallow water flow. The novel contribution of this study consists in designing a finite-difference weighted essentially non-oscillatory (WENO) scheme based on the alternative formulation of the shallow water flow model, denoted as "pre-balanced" shallow water equations and introduced in [Journal of Computational Physics 192 (2003) 422-451]. This formulation greatly simplifies the achievement of the well-balancing of the present scheme. Rigorous numerical analysis, as well as extensive numerical results all, verify that the current scheme preserves the exact conservation property. It is important to note that this resulting scheme also maintains the non-oscillatory property near discontinuities and keep high-order accuracy for smooth solutions at the same time.