A high-order spectral (finite) volume method for conservation laws on unstructured grids

Abstract

A time accurate, high-order, conservative, finite method named Spectral Volume (SV) method has been developed for conservation laws on unstructured grids. The concept of a volume is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multi-domain spectral methods. Each spectral is partitioned into control volumes (CVs), and cell-averaged data from these control volumes is used to reconstruct a high-order polynomial approximation in the spectral volume. Riemann solvers are used to compute the fluxes at spectral boundaries. Then cell-averaged state variables in the control volumes are updated independently. Furthermore, TVD (Total Variation Diminishing) and TVB (Total Variation Bounded) limiters are introduced in the SV method to remove spurious oscillations near discontinuities. A very desirable feature of the SV method is that the reconstruction is identical for cells of the same type with similar partitions, and that the reconstruction stencil is always non-singular, in contrast to the memory and CPU-intensive reconstruction in a high-order k-exact finite (FV) method. The high-order accuracy of the SV method is demonstrated for several model problems with and without discontinuities.