A High-Resolution Rotated Grid Method for the Approximation of Conservation Laws in Complex Geometries

We consider the approximation of multidimensional systems of conservation laws on Cartesian grids with embedded irregular boundaries. Our aim is to obtain a stable and accurate approximation with explicit finite volume methods using time steps that are appropriate for the regular grid cells. Grid cells near the boundary may be orders of magnitude smaller than a regular grid cell. Our approach is based on the so-called h-box method of Berger and LeVeque. This Godunov-type method calculates fluxes at cell interfaces by solving Riemann problems defined over boxes of a reference grid cell length h, i.e. the length of a regular grid cell. It turns out that the accuracy of an h-box method depends strongly on the definition of h-box values. We present a new second order accurate h-box method for the approximation of conservation laws on one-dimensional irregular grids. We use insight obtained from this test problem to construct high-resolution rotated grid methods that can handle embedded boundaries in two-dimensional calculations.