Positivity-preserving schemes in multidimensions (Q2706433)

The performance of the MUSCL-Hancock upwind scheme is examined on problems of two-dimensional advection. It is found that when the advection direction is skewed relative to the mesh, most discrete total variation diminishing (TVD) limiters give rise to large spurious oscillations near discontinuities. The cause of these oscillations is traced to reconstructions that are not bounded by neighboring cell averages. It is proved that if the reconstruction in each cell is bounded by the cell averages of first order neighbors, then the MUSCL-Hancock scheme is positivity preserving. A simple limiter that achieves such bounded reconstructions is presented next along with a variant that is uniformly second order accurate. Numerical experiments show that the new schemes are accurate and efficient and compare favorably with other schemes.