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Volume 42, Issue 6
Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients

Minqiang Xu, Yanting Yuan, Waixiang Cao & Qingsong Zou

J. Comp. Math., 42 (2024), pp. 1627-1655.

Published online: 2024-11

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  • 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.

  • AMS Subject Headings

65N30, 65N25, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-1627, author = {Xu , MinqiangYuan , YantingCao , Waixiang 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 = {

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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2305-m2021-0330}, url = {http://global-sci.org/intro/article_detail/jcm/23510.html} }
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 -

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.

Xu , MinqiangYuan , YantingCao , Waixiang and Zou , Qingsong. (2024). Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients. Journal of Computational Mathematics. 42 (6). 1627-1655. doi:10.4208/jcm.2305-m2021-0330
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