Error estimates of a regularized finite difference method for the logarithmic Schr\"odinger equation

TL;DR

This paper introduces a regularized finite difference method for the logarithmic Schr"odinger equation, providing error bounds and numerical validation despite the challenges posed by the nonlinearity's blow-up behavior.

Contribution

It proposes a regularized model for LogSE and develops a semi-implicit finite difference scheme with proven error estimates, addressing numerical difficulties from the nonlinearity.

Findings

Error bounds established for the regularized scheme.

Numerical results confirm theoretical error estimates.

Linear convergence between RLogSE and LogSE solutions.

Abstract

We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$ with $\rho=|u|^2$ 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 round-off error and to avoid blow-up, a regularized logarithmic Schr\"odinger equation (RLogSE) is proposed with a small regularization parameter $0<\varepsilon\ll 1$ and linear convergence is established between the solutions of RLogSE and LogSE in term of $\varepsilon$. 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 $\varepsilon$. Finally numerical results are reported to confirm our error bounds.