Multidimensional Upwind Methods for Hyperbolic Conservation Laws

We present a class of second-order conservative finite difference algorithms for solving numerically time-dependent problems for hyperbolic conservation laws in several space variables. These methods are upwind and multidimensional, in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge, and that all fluxes are evaluated and differenced at the same time; in particular, operator splitting is not used. Correct behavior at discontinuities is obtained by the use of solutions to the Riemann problem, and by limiting some of the second-order terms. Numerical results are presented, which show that the methods described here yield the same high resolution as the corresponding operator split methods.