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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) nite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L2 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.

Optimal error estimates to smooth solutions of the central discontinuous Galerkin methods for nonlinear scalar conservation laws