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Analysis of Two Any Order Spectral Volume Methods for 1-D Linear Hyperbolic Equations with Degenerate Variable Coefficients
Minqiang Xu, Yanting Yuan, Waixiang Cao and Qingsong Zou

J. Comp. Math. DOI: 10.4208/jcm.2305-m2021-0330

Publication Date : 2023-11-27

  • 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.

  • Copyright

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