Superconvergence analysis of local discontinuous Galerkin methods for linear convection–diffusion equations in one space dimension

Abstract

This paper is concerned with the superconvergence study of the local discontinuous Galerkin (LDG) method for one-dimensional time-dependent linear convection–diffusion equations, where the convection flux is taken as the upwind flux, while the diffusion fluxes chosen as the alternating fluxes. Superconvergence properties for both the solution itself and auxiliary variables are established. Precisely, we prove that, the LDG solutions are superconvergent with an order of $k+2$ towards a particular projection of the exact solution and the auxiliary variable, and thus a $k+1$-th order superconvergence for the derivative approximation and a $k+2$-th order superconvergence for the function value approximation at a class of Radau points are obtained. Especially, we show that the convergence rate of the derivative approximation for the exact solution can reach $k+2$ when the convection flux is the same as the diffusion flux, two order higher than the optimal convergence rate. Furthermore, a $2k+1$-th order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages, is also obtained under some suitable initial discretization. Numerical experiments indicate that most of our theoretical findings are optimal.