ADER DG and FV Schemes for Shallow Water Flows

Abstract

We are concern with ADER [3] high-order numerical methods for the timedependent two-dimensional non-linear shallow water equations [2] in the framework of finite volumes (FV) and discontinuous Galerkin (DG) finite elements methods using non-structured triangular meshes. High order in space and time is obtained by (1) a high-order spatial distribution of the solution in each element, (2) the solution of the derivative Riemann problem (DRP) [4] and (3) an accurate computation of the numerical flux and volume integrals. Regarding the high-order spatial distribution of the solution, in the FV method one requires cell average reconstructions [1] at each time step; in the case of DG the high-order representation of the data is built into the scheme to the desired order and no reconstruction is needed. However, in the presence of high gradients numerical oscillations arise in the DG case, which requires the implementation of special reconstruction. We assess the methods by comparing numerical solutions with exact solutions. The rest of the chapter is organized as follows: Sect. 2 describes the governing equations and the hyperbolic character. In Sect. 3 we construct the numerical method. In Sect. 4 we describe the ADER approach. Test problems are presented in Sect. 5 and conclusions are drawn in Sect. 6.