Abstract

Shallow water moment models constitute an extension of the standard shallow water equations in which the velocity field is described by a polynomial expansion with respect to the vertical coordinate. In addition to the water thickness and depth-averaged velocity, these free-surface flow models introduce the coefficients of the polynomial velocity expansion, commonly referred to as moments, as additional unknowns. The corresponding moment equations are derived through Galerkin projection.

In this presentation, we review existing shallow water moment models and discuss their main properties. We then derive the associated energy equations and entropy variables. Based on the nodal discontinuous Galerkin spectral element method, we develop a high-order entropy-preserving scheme for the approximation of these models. To this end, we construct numerical fluxes following the continuous entropy analysis and designed to satisfy an entropy-preserving condition in the presence of non-conservative terms. The resulting scheme is implemented in Julia using the Trixi framework. Numerical experiments demonstrate the robustness of the method and confirm entropy preservation as well as the expected order of convergence.