A Large Time Step explicit scheme (CFL>1) on unstructured grids for 2D conservation laws: Application to the homogeneous shallow water equations

Abstract

• A new explicit technique to exceed the CFL = 1 limit in 2D simulations is proposed. • It is specifically designed for unstructured triangular flexible meshes. • The method is suitable for scalar and systems of conservation laws. • It is applied to the 2D homogeneous shallow water equations. In this work, a Large Time Step (LTS) explicit finite volume scheme designed to allow CFL > 1 is applied to the numerical resolution of 2D scalar and systems of conservation laws on triangular grids. Based on the flux difference splitting formulation, a special concern is put on finding the way of packing the information to compute the numerical solution when working on unstructured grids. Not only the cell areas but also the length of the interfaces and their orientation are questions of interest to send the information from each edge or interface. The information to update the cell variables is computed according to the local average discrete velocity and the orientation of the edges of the cells involved. The performance of these ideas is tested and compared with the conventional explicit first order and second order schemes in academic configurations for the 2D linear scalar equation and for 2D systems of conservation laws (in particular the shallow water equations) without source terms. The LTS scheme is demonstrated to preserve or even gain accuracy and save computational time with respect to the first order scheme.