Error estimates of local energy regularization for the logarithmic Schrodinger equation

The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications. Due to the singularity of the logarithmic function, it introduces tremendous difficulties in establishing mathematical theories, as well as in designing and analyzing numerical methods for PDEs with such nonlinearity. Here, we take the logarithmic Schrodinger equation (LogSE) as a prototype model. Instead of regularizing f(rho) =ln rho in the LogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE by first regularizing F(rho) = rho ln rho - rho locally near rho = 0+ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrodinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter 0 < epsilon << 1. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improves the linear convergence rate of the regularization method in the literature. Error estimates are also presented for solving the ERLogSE by using Lie-Trotter splitting integrators. Numerical results are reported to confirm our error estimates of the LER and of the time-splitting integrators for the ERLogSE. Finally, our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.