Definition of the weak solution for the 2D shallow water equations with source terms in presence of source terms using Roe's approach

摘要

Actually, Roe's approach is one of the most used weak solutions involved in the context of numerical simulation of free surface flows. The Godunov method formulates an approximate solution of the PDE between two numerical cells solving a Riemann problem. Roe's method can be applied successfully if avoiding source terms, but when included instabilities arise. An upwind discretization of the source term enforcing equilibrium in the cases of still water has been found as a reliable option, but erroneous results can still appear. Here, the definition of the weak solutions for the 1D and 2D shallow water equations are presented. It is shown that the definition of well-balanced equilibrium in trivial cases is not sufficient. These conclusions will be proved comparing the numerical solutions with exact solutions in 2D Riemann problems with source terms.