@Article{JCM-42-6,
author = {Xu, Minqiang and Yuan, Yanting and Waixiang, Cao and Zou, Qingsong},
title = {Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients},
journal = {Journal of Computational Mathematics},
year = {2024},
volume = {42},
number = {6},
pages = {1627--1655},
abstract = {<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>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2305-m2021-0330},
url = {https://global-sci.com/article/91411/analysis-of-two-any-order-spectral-volume-methods-for-1-d-linear-hyperbolic-equations-with-degenerate-variable-coefficients}
}