TY - JOUR
T1 - Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients
AU - Xu , Minqiang
AU - Yuan , Yanting
AU - Cao , Waixiang
AU - Zou , Qingsong
JO - Journal of Computational Mathematics
VL - 6
SP - 1627
EP - 1655
PY - 2024
DA - 2024/11
SN - 42
DO - http://doi.org/10.4208/jcm.2305-m2021-0330
UR - https://global-sci.org/intro/article_detail/jcm/23510.html
KW - Spectral Volume Methods, $L^2$ stability, Error estimates, Superconvergence.
AB - <p style="text-align: justify;">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.</p>