Two types of spectral volume methods for 1-D linear hyperbolic equations with degenerate variable coefficients

TL;DR

This paper introduces and analyzes two spectral volume methods for 1-D hyperbolic equations with variable coefficients, demonstrating their stability, optimal convergence, and superconvergence properties through rigorous proofs and numerical validation.

Contribution

It develops two classes of spectral volume methods for degenerate hyperbolic equations, proving their stability, convergence, and superconvergence, which advances numerical solutions for such equations.

Findings

Both methods are $L^2$-norm stable on non-uniform meshes.

They achieve optimal order convergence.

Superconvergence occurs at order $(k+2)$ for flux and $(k+1.5)$ for solutions.

Abstract

In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge 1$ is an arbitrary integer) polynomial function satisfy the local conservation law in each {\it control volume} obtained by dividing the interval element of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radaus points (RSV). The $L^2$-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes. The superconvergence behaviors of the two SV schemes have been also investigated: it is proved that under the $L^2$ norm, the SV flux function approximates the exact flux with $(k+2)$-th order and the SV solution approximates the exact solution with $(k+\frac32)$-th order; some superconvergence behaviors at certain special points and for element averages have been also discovered and proved. Our theoretical findings are verified by several numerical experiments.