ON THE CONVERGENCE OF THE CRANK-NICOLSON METHOD FOR THE LOGARITHMIC SCHRODINGER EQUATION

We consider an initial and Dirichlet boundary value problem for a logarithmic Schrodinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete L-t(infinity) (L-x(2)) norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrodinger equation is provided, without regularizing its nonlinear term.