Solving the nonlinear Schrödinger equation using exponential integrators

Using the notion of integrating factors, Lawson developed a class of numerical methods for solving stiff systems of ordinary differential equations. However, the performance of these "Generalized Runge - Kutta processes" was demonstrably poorer when compared to the ETD schemes of Certaine and Nørsett, recently rediscovered by Cox and Matthews. The deficit is particularly pronounced when the schemes are applied to parabolic problems. In this paper we compare a fourth order Lawson scheme and a fourth order ETD scheme due to Cox and Matthews, using the nonlinear Schro¨dinger equation as the test problem. The primary testing parameters are degree of regularity of the potential function and the initial condition, and numerical performance is heavily dependent upon these values. The Lawson and ETD schemes exhibit significant performance differences in our tests, and we present some analysis on this.