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

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