2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

Abstract

In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution.