Superconvergence of Local Discontinuous Galerkin methods for one-dimensional linear parabolic equations

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrodinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the \(L^{2}\) norm. The order of superconvergence is \(2k+1\), when the polynomials of degree at most k are used. Even though the linear Schrodinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a \((2k+1)\)-th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of \((k+2)\)-th order at the Radau points. All theoretical findings are confirmed by numerical experiments.