Stability analysis of the cell centered finite-volume MUSCL method on unstructured grids

The goal of this study is to apply the MUSCL scheme to the linear advection equation on general unstructured grids and to examine the eigenvalue stability of the resulting linear semi-discrete equation. Although this semi-discrete scheme is in general stable on cartesian grids, numerical calculations of spectra show that this can sometimes fail for generalizations of the MUSCL method to unstructured three-dimensional grids. This motivates our investigation of the influence of the slope reconstruction method and stencil on the eigenvalue stability of the MUSCL scheme. A theoretical stability analysis of the first order upwind scheme proves that this method is stable on arbitrary grids. In contrast, a general theoretical result is very difficult to obtain for the MUSCL scheme. We are able to identify a local property of the slope reconstruction that is strongly related to the appearance of unstable eigenmodes. This property allows to identify the reconstruction methods that are best suited for stable discretizations. The explicit numerical computation of spectra for a large number of two-and three-dimensional test cases confirms and completes the theoretical results.