Two types of spectral volume methods for 1-D linear hyperbolic equations with degenerate variable coefficients
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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≥ 1 is an arbitrary integer) polynomial function satisfy the local conservation law in each 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+3/2)-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.
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