On a class of genuinely 2D incomplete Riemann solvers for hyperbolic systems

Abstract

We propose a general class of genuinely two-dimensional (2D) incomplete Riemann solvers for hyperbolic systems of equations. In particular, extensions of the multidimensional HLL solver proposed by Balsara to 2D Polynomial Viscosity Matrix (PVM) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM flux with two purely 2D PVM fluxes defined at corners, which take into account transversal features of the flow through the approximate solution of 2D Riemann problems. The proposed methods are applicable to general hyperbolic systems of conservation laws. A first version of the scheme was recently introduced in our other work, where applications to magnetohydrodynamics were considered, including an efficient technique for divergence cleaning of the magnetic field based on the nonconservative form of the ideal magnetohydrodynamics equations, which provides good results in combination with our 2D solvers. In this work, we focus on applications to shallow water systems, in which the source term due to the bottom topography introduces an additional difficulty. An elegant way to overcome this difficulty consists in reformulating the problem in nonconservative form. For this reason, we have extended our 2D schemes to the case of nonconservative hyperbolic systems, within the framework of path-conservative schemes introduced in the work of Parés. Again, the proposed schemes are applicable to general nonconservative hyperbolic systems. A number of challenging numerical experiments including genuinely 2D effects are presented to test the performances and advantages of the proposed schemes.