High order finite difference WENO schemes with the exact conservation property for the shallow water equations

Abstract

Shallow water equations with nonflat bottom have steady state solutions in which the flux gradients are nonzero but exactly balanced by the source term. It is a challenge to design genuinely high order accurate numerical schemes which preserve exactly these steady state solutions. In this paper we design high order finite difference WENO schemes to this system with such exact conservation property (C-property) and at the same time maintaining genuine high order accuracy. Extensive one and two dimensional simulations are performed to verify high order accuracy, the exact C-property, and good resolution for smooth and discontinuous solutions.