Analysis of Spectral Volume Methods for 1D Linear Scalar Hyperbolic Equations

This paper is concerned with the analysis of two spectral volume (SV) methods for 1D scalar hyperbolic equations : one is constructed basing on the Gauss-Legendre points (LSV) and the other is based on the right-Radau points (RRSV). We first prove that for a general nonuniform mesh and any polynomial degree k, both the LSV and RRSV methods are stable and can achieve optimal convergence orders in the L-2 norm. Secondly, we prove that both methods have some superconvergence properties at some special points. For instances, at the downwind points, the solution of RRSV and LSV methods converges with the order of O(h(2k+1)) and O(h(2k)), respectively. Moreover, we demonstrate that for constant-coefficient equations, the RRSVmethod is identical to the upwind discontinuous Galerkin (DG) method. Our theoretical findings are validated with several numerical experiments at the end.