On a class of two-dimensional incomplete Riemann solvers

We propose a general class of genuinely two-dimensional incomplete Riemann solvers for systems of conservation laws. In particular, extensions of Balsara's multidimensional HLL scheme [J. Comput. Phys. 231 (2012) 7476-7503] to two-dimensional PVM/RVM (Polynomial/Rational Viscosity Matrix) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM/RVM flux with two purely two-dimensional PVM/RVM fluxes at vertices, which take into account transversal features of the flow. The proposed methods are applicable to general hyperbolic systems, although in this paper we focus on applications to magnetohydrodynamics. In particular, we propose an efficient technique for divergence cleaning of the magnetic field that provides good results when combined with our two-dimensional solvers. Several numerical tests including genuinely two-dimensional effects are presented to test the performances of the proposed schemes.