Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients (Journal of Computational Mathematics, 2024)
In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. Two classes of SV methods are constructed by letting a piecewise $k$-th order ($k ≥ 1$ is an integer) polynomial tosatisfy the conservation law in each control volume, which is obtained by refining spectral volumes (SV) of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radauspoints (RSV) in each SV. The $L^2$-norm stability and optimal order convergence propertiesfor both methods are rigorously proved for general non-uniform meshes. Surprisingly, wediscover some very interesting superconvergence phenomena: At some special points, theSV flux function approximates the exact flux with $(k+2)$-th order and the SV solution itselfapproximates the exact solution with $(k+3/2)$-th order, some superconvergence behaviorsfor element averages errors have been also discovered. Moreover, these superconvergencephenomena are rigorously proved by using the so-called correction function method. Ourtheoretical findings are verified by several numerical experiments.
Keywords: Spectral Volume Methods; $L^2$ stability; Error estimates; Superconvergence.