Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation
This work provides rigorous error analysis for a numerical method addressing the blow-up issue in the logarithmic Schrödinger equation, benefiting computational scientists studying nonlinear wave phenomena.
The authors propose a regularized finite difference method for the logarithmic Schrödinger equation, proving linear convergence in the regularization parameter and error bounds in mesh size and time step, confirmed by numerical tests.
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. $\ln ρ\to -\infty$ when $ρ\rightarrow 0^+$ with $ρ=|u|^2$ 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 $0<\varepsilon\ll 1$ 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 $τ$ as well as the small regularization parameter $\varepsilon$. Finally numerical results are reported to confirm our error bounds.