Abstract: We study a generalization 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, accuracy and entropy consistency 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, that is based on the multidimensional wave propagation algorithm, is also described.
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