Error analysis of a first-order IMEX scheme for the logarithmic Schr\"odinger equation

TL;DR

This paper analyzes the error of a first-order IMEX scheme for the logarithmic Schrödinger equation, addressing challenges posed by the non-differentiable logarithmic nonlinearity and providing numerical validation of convergence.

Contribution

It is the first to study a direct linearized IMEX scheme for the LogSE, introducing new analytical tools for error analysis in this context.

Findings

Error bounds for the IMEX scheme are established.

Numerical results confirm the expected convergence rates.

New tools include H"older continuity characterization and nonlinear Gr"onwall's inequality.

Abstract

The logarithmic Schr\"odinger equation (LogSE) has a logarithmic nonlinearity $f(u)=u\ln |u|^2$ that is not differentiable at $u=0.$ Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularising $f(u)$ as $ u^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^2$ to overcome the blowup of $\ln |u|^2$ at $u=0$ has been investigated recently in literature. With the understanding of $f(0)=0,$ we analyze the non-regularized first-order Implicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the H\"older continuity of the logarithmic term, and a nonlinear Gr\"{o}nwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first one to study the direct linearized scheme for the LogSE as far as we can tell.