摘要
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 to satisfy 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 Radaus points(RSV)in each SV.The L^(2)-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes.Surprisingly,we discover some very interesting superconvergence phenomena:At some special points,the SV flux function approximates the exact flux with(k+2)-th order and the SV solution itself approximates the exact solution with(k+3/2)-th order,some superconvergence behaviors for element averages errors have been also discovered.Moreover,these superconvergence phenomena are rigorously proved by using the so-called correction function method.Our theoretical findings are verified by several numerical experiments.
基金
supported by the NSFC(Grants 92370113,12071496,12271482)
Moreover,the first author was also supported by the Zhejiang Provincial NSF(Grant LZ23A010006)
by the Key Research Project of Zhejiang Lab(Grant 2022PE0AC01)
the fourth author was also supported by the Guangdong Provincial NSF(Grant 2023A1515012097).
