Anti-diffusive, Non-oscillatory Central (adNOC) scheme to solve the shallow water equations over an erodible substrate in two dimensions

Abstract

Shallow water surface flows commonly entrain sediments, resulting in scouring and/or deposition of the underlying substrate that may strongly influence the pattern of subsequent flow. These coupled phenomena, which can be investigated mathematically with some extension of the shallow water equations, present numerous challenges for numerical methods. Here, we present a straightforward Riemann-solver free approach to solve these equations based on the explicit non-oscillatory central (NOC) scheme that has already been widely applied to hyperbolic conservation laws in other contexts. Our version of the central scheme is second-order accurate in time and space and is used with an anti-diffusive correction to reduce numerical diffusion usually suffered by central schemes. Numerical experiments show that the scheme is accurate and robust for a range of applications from highly dynamic spontaneous dam break over a mobile bed to slowly evolving morphological bed in an alluvial river.