Abstract
We 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.