Recent developments in modeling free-surface flows with vertically-resolved velocity profiles using moments

Abstract

Shallow water moment models are non-linear PDEs in balance law form for free-surface flows that allow for vertical variations in the horizontal velocity. The models are extensions of the standard shallow water equations. However, the models in their original form lack global hyperbolicity. The loss of hyperbolicity already occurs for small vertical variations of the velocity and this leads to instabilities in numerical test cases. We review two recently developed hyperbolic shallow water moment models, which are based on two different linearizations during the derivation. Recently, the models have been extended to consider sediment transport and bottom topographies, for which new well-balanced numerical schemes based on analytical derivation of steady states can be constructed. We summarize the recent developments focusing on analytical properties of the models and their derivation.