Numerical stability of explicit Runge-Kutta finite-difference schemes for the nonlinear Schrodinger equation

Linearized numerical stability bounds for solving the nonlinear time-dependent Schrodinger equation (NLSE) using explicit finite-differencing are shown. The bounds are computed for the fourth-order Runge-Kutta scheme in time and both second-order and fourth-order central differencing in space. Results are given for Dirichlet, modulus-squared Dirichlet, Laplacian-zero, and periodic boundary conditions for one, two, and three dimensions. Our approach is to use standard Runge-Kutta linear stability theory, treating the nonlinearity of the NLSE as a constant. The required bounds on the eigenvalues of the scheme matrices are found analytically when possible, and otherwise estimated using the Gershgorin circle theorem. (C) 2013 IMACS. Published by Elsevier B.V. All rights reserved.