Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients
Year: 2024
Author: Minqiang Xu, Yanting Yuan, Waixiang Cao, Qingsong Zou
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 6 : pp. 1627–1655
Abstract
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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2305-m2021-0330
Journal of Computational Mathematics, Vol. 42 (2024), Iss. 6 : pp. 1627–1655
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Spectral Volume Methods $L^2$ stability Error estimates Superconvergence.
Author Details
Minqiang Xu Email
Yanting Yuan Email
Waixiang Cao Email
Qingsong Zou Email