A 3rd/2nd order MOOD limited scheme for the shallow water equations

Abstract

AbstractWe present a high‐order accurate, positivity‐preserving and well‐balanced finite volume scheme for the shallow water equations with variable topography. An unlimited third‐order scheme is combined with the recent, second‐order accurate Bottom‐Surface‐Gradient Method (BSGM, [5]). This is monitored by an a‐posteriori MOOD (Multidimensional Optimal Order Detection) limiting step [2, 7–9], which detects possible local instabilities of a high‐order candidate solution such as loss of positivity or local oscillations, and switches locally to a lower order, stable and robust “parachute” scheme if necessary. We demonstrate the accuracy, effectiveness and robustness of the proposed adaptive methodology with numerical experiments, both for near‐equilibrium and non‐equilibrium depth‐averaged flows.