H-Box Methods for the Approximation of Hyperbolic Conservation Laws on Irregular Grids

We study generalizations of the high-resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization. This Godunov-type scheme calculates fluxes at cell interfaces by solving Riemann problems defined over boxes of a reference grid cell length h. We discuss stability and accuracy of the resulting so-called h-box methods for one-dimensional systems of conservation laws. An extension of the method for the two-dimensional case, which is based on the multidimensional wave propagation algorithm, is also described.