On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

Abstract

<p style='text-indent:20px;'>We consider an initial and Dirichlet boundary value problem for a logarithmic Schrödinger equation over a two dimensional rectangular domain. We construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization, with or without regularizing the logarithmic term. We develop a convergence analysis yielding a new almost second order a priori error estimates in the discrete <inline-formula><tex-math id="M1">\begin{document}$ L_t^{\infty}(L_x^2) $\end{document}</tex-math></inline-formula> norm, and we show results from numerical experiments exposing the efficiency of the method proposed. It is the first time in the literature where an error estimate for a numerical method applied to the logarithmic Schrödinger equation is provided, without regularizing its nonlinear term.</p>

History of Mathematics in American Mathematical Textbooks of the Late 19th and Early 20th Centuries

“…Using [1, Lemma 3.1] and following the argument in [28] and [5, sec. 2], we can show the unique solvability of (3.1) (see Appendix A for the sketch of the proof).…”

“…With this understanding, we propose to directly discretize the original LogSE (1.1) without regularizing the nonlinear term. In the course of finalising this work, we realized that Paraschis and Zouraris [28] analysed the (implicit) Crank-Nicolson scheme for (1.1) (without regularization), so it required solving a nonlinear system at each time step. Note that the fixed-point iteration was employed therein as the non-differentiable logarithmic nonlinear term ruled out the use of Newton-type iterative methods.…”