A RKDG scheme for 2D SWE on curved boundary domains

This work regards a high-order numerical scheme for the integration of the two-dimensional Shallow Water Equations (SWE). First, we highlight the importance of the high-order representation of the boundaries when the numerical scheme has a high-order accuracy. We show the impossibility to obtain a physically realistic solution for some families of steady-state problems using only straight-sided elements. Moreover, we prove that the existing techniques are not suitable to obtain a well-balanced model for curved boundaries. Our solution to the examined problems consists of a third-order-accurate, well-balanced, Runge-Kutta discontinuous Galerkin (RKDG) model for the integration of the SWE on an unstructured triangular grid. The problem of the curved boundary is addressed by using a proper mixture of straight-sided elements, in the inner part of the computational domain, and of elements with a single curved edge, in the regions near the boundaries. A careful combination of available techniques yields a well-balanced model on the straight-sided elements, and an original approach is proposed for balancing the model on the curved-sided elements. This approach is based on a modified mathematical model of SWE which is consistent with the original one and allows to achieve the well-balancing property.