A Conservative Formulation of the Multidimensional Upwind Residual Distribution Schemes for General Nonlinear Conservation Laws

In the present paper we consider the numerical solution of systems of general nonlinear hyperbolic conservation laws on unstructured grids by means of the residual distribution method. We propose a new formulation of the first-order linear, optimal positive N scheme, relying on a contour integration of the convective fluxes over the boundaries of an element. Full conservation is achieved for arbitrary flux functions, while the robustness and the monotone shock capturing of the original N scheme is retained. The new variant of the N scheme is combined with the conservative second-order linear LDA scheme to obtain a nonlinear second-order monotone B scheme. The performance of the new residual distribution schemes is evaluated on problems governed by the Euler equations. As an application to a more complex system of conservation laws lacking an exact conservative linearization, we solve the ideal magnetohydrodynamics equations in two spatial dimensions.