A well-balanced, third-order-accurate RKDG scheme for SWE on curved boundary domains

In this work, we present a third-order-accurate Runge–Kutta discontinuous Galerkin model defined on an unstructured triangular grid. The originality of this model lies in its ability to correctly and efficiently treat computational domains with curved boundaries while preserving the exact well-balancing between the source terms and the flux divergence in the case of quiescent flow. With this aim, the model uses straight-sided elements in the inner part of the domain and curved-sided elements in the area adjacent to the boundaries, limiting as much as possible the additional computational costs due to the use of high-order isoparametric elements. Together with a careful combination of well-known techniques used for the exact balancing of the scheme with straight-sided elements, an original solution is proposed to achieve the same balancing for the curved-sided elements. Such an approach is based on a locally modified formulation of the shallow water equations. Proofs of the consistency between the modified formulation and the original one, of the satisfaction of the C-property for quiescent flow and of the negligibility of the added new terms in the case of moving water are provided. Several examples are presented to demonstrate the well-balancing property and the overall good behavior of the proposed scheme.