Numerical solution of the regularized logarithmic Schrodinger equation on unbounded domains

The numerical solution of the logarithmic Schrodinger equation on unbounded domains is considered in this paper. It is difficult to develop numerical methods for the logarithmic Schrodinger equation on unbounded domains, due to the blow up of the logarithmic nonlinearity and the unboundedness of the physical domain. Thus, a regularized version of the logarithmic Schrodinger equation on unbounded domains with a small regularization parameter is developed. Then, the local artificial boundary conditions for the regularized logarithmic Schrodinger equation are designed by applying the unified approach, which based on the idea of well-known operator splitting method. The regularized logarithmic Schrodinger equation defined on unbounded domains is reduced to an initial boundary value problem on the bounded computational domain, which can be solved by the finite difference method. The convergence and the stability of the reduced problem are analyzed by introducing some auxiliary variables. In order to choose the optimal absorb parameter in the local artificial boundary conditions, an adaptive algorithm is presented. Numerical results are reported to verify the accuracy and effectiveness of our proposed method. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.