Applied Analysis and Computational Mathematics

Many second-order accurate finite volume methods are based on the following 3 steps: 1. Construct a (limited) linear polynomial in every cell out of the given cell averages (unij). 2. Calculate the fluxes through the cell edges by solving a Riemann problem using states based on the linear polynomial. 3. Update the cell averages to (un+1ij) using the fluxes through the edges. In Step 1, a limiter is needed which adjusts the slope appropriately. The limiter prevents overshoots and oscillations that arise at discontinuities, as well as maintaining positivity, since for example if pressure becomes negative your code will crash. Limiters in one dimension are very well understood. In two dimensions for rectangular grids a one-dimensional limiter is usually applied in the x and y direction separately. This approach is very straightforward and usually works well. However, this splitting of x and y components of the gradient relies on the cell edges being aligned with the coordinate axes. On non-coordinate-aligned grids a two-dimensional limiter is much more difficult. This is es-pecially true for embedded boundary grids with ‘cut cells’, the motivation for this work. Thes