Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation

Abstract

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

23 pages, 4 figures

References in corpus (2)

• Universal dynamics for the defocusing logarithmic Schrodinger equation
• A note on the nonlinear Schrödinger equation in a general domain

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