Weak solutions for partial differential equations with source terms: Application to the shallow water equations

Abstract

Weak solutions of problems with m equations with source terms are proposed using an augmented Riemann solver defined by m+1 states instead of increasing the number of involved equations. These weak solutions use propagating jump discontinuities connecting the m+1 states to approximate the Riemann solution. The average of the propagated waves in the computational cell leads to a reinterpretation of the Roe's approach and in the upwind treatment of the source term of Vazquez-Cendon. It is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities, as usually done in the homogeneous case. Positivity requirements over the values of the intermediate states are the only way to control the global stability of the method. Also it is shown that the definition of well-balanced equilibrium in trivial cases is not sufficient to provide correct results: it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded. The one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study. The stability region is shown to differ from the one defined for the case without source terms, and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet/dry front is a particular case, of the wet/wet fronts. The consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case. Starting from this result, 1D and 2D numerical schemes are developed for both quadrilateral and triangular grids, enforcing conservation and positivity over the solution, allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size.