Partition Design and Optimization for High Order Spectral Volume Schemes

Abstract

An analysis of the accuracy and stability properties of the spectral volume (SV) method, with applicability to very high-order accurate simulations, is presented. In the SV method, each simplex grid cell is called a spectral volume (SV), and the SV is further partitioned into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In general, the partitioning of an SV into CVs is not uniquely defined, and thus it is of great importance to select a partition which yields favorable stability properties, and results in an interpolation polynomial of high quality. Here we present a new approach to efficiently locate stable partitions by means of constrained minimization. This is motivated by the fact that, at present, an exhaustive search approach to SV partition design would be prohibitively costly and thus not feasible. Once stable partitions are located, a high quality interpolation polynomial is then assured by subsequently minimizing the dissipation and dispersion errors of the stable partitions. Results are presented which demonstrate the potential of this method for producing stable and highly accurate partitions of arbitrary order. In particular, a new 4 th -order partition is presented which has improved accuracy and stability properties over previously used partitions, and a new stable 5 th -order partition is introduced.