Error estimates of local energy regularization for the logarithmic Schrodinger equation

Abstract

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