Superconvergence analysis of spectral volume methods for one-dimensional diffusion and third-order wave equations

We present a unified approach to studying the superconvergence property of the spectral volume (SV) method for high-order time-dependent partial differential equations using the local discontinuous Galerkin formulation. We choose the diffusion and third-order wave equations as our models to illustrate approach and the main idea. The SV scheme is designed with control volumes constructed using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as GSV and RSV schemes, respectively. With a careful choice of numerical fluxes, we demonstrate that the schemes are stable and exhibit optimal error estimates. Furthermore, we establish superconvergence of the GSV and RSV for the solution itself and the auxiliary variables. To be more precise, we prove that the errors of numerical fluxes at nodes and for the cell averages are superconvergent with orders of O(h2k+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{O}(h<^>{2k+1})$$\end{document} and O(h2k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{O}(h<^>{2k})$$\end{document} for RSV and GSV, respectively. Superconvergence for the function value and derivative value approximations is also studied and the superconvergence points are identified at Gauss points and Radau points. Numerical experiments are presented to illustrate theoretical findings.