Superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional linearized KdV equations

In this paper, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes to study the superconvergent properties of one-dimensional linearized KdV equations. Compared with traditional upwind and alternating fluxes, a slower error growth of the LDG solution using generalized numerical fluxes can be obtained for long time simulations. By establishing five energy identities and properties of correction functions with the appropriate numerical initial condition, we derive the supercloseness between the LDG solution and the interpolation function. The errors of the numerical fluxes as well as the cell averages achieve the $(2k+1)$th-order superconvergence. In addition, we prove that the superconvergent rates of the function and derivative values at the interior generalized Radau points are $k+2$ and $k+1$, respectively. An extension to mixed boundary conditions is given, for which we present the generalized skew-symmetry property and propose an appropriate conservation property for the numerical initial condition. Numerical experiments are shown to demonstrate the theoretical results, including cases with other boundary conditions and nonlinear KdV equations.