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 introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst 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 Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<\ep\ll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.