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Abstract In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results. 

In this paper, we study the error estimates to sufficiently smooth solutions of the nonlinear scalar conservation laws for the semi-discrete central discontinuous Galerkin (DG) finite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L 2 error estimates of the semi-discrete CDG schemes, and this condition is checked to be valid in one and two dimensions for polynomials of degree up to k = 8. Numerical experiments are given to verify the theoretical results.