Two types of spectral volume methods for 1-D linear hyperbolic equations with degenerate variable coefficients

Abstract

In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge 1$ is an arbitrary integer) polynomial function satisfy the local conservation law in each {\it control volume} obtained by dividing the interval element of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radaus points (RSV). The $L^2$-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes. The superconvergence behaviors of the two SV schemes have been also investigated: it is proved that under the $L^2$ norm, the SV flux function approximates the exact flux with $(k+2)$-th order and the SV solution approximates the exact solution with $(k+\frac{3}{2})$-th order; some superconvergence behaviors at certain special points and for element averages have been also discovered and proved. Our theoretical findings are verified by several numerical experiments.