ERROR ESTIMATES OF A REGULARIZED FINITE DIFFERENCE METHOD FOR THE LOGARITHMIC SCHRODINGER EQUATION

摘要

We present a regularized finite difference method for the logarithmic Schrodinger equation (LogSE) and establish its error bound. Due to the blowup of the logarithmic nonlinearity, i.e., ln rho -> -infinity when rho -> 0+ with rho = vertical bar u vertical bar(2) being the density and u being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the roundoff error and to avoid blowup, a regularized LogSE (RLogSE) is proposed with a small regularization parameter 0 < epsilon 1 and linear convergence is established between the solutions of RLogSE and LogSE in term of epsilon. Then a semi-implicit finite difference method is presented for discretizing the RLogSE and error estimates are established in terms of the mesh size h and time step tau as well as the small regularization parameter epsilon. Finally numerical results are reported to illustrate our error bounds.