Comparison of adaptive step-size control strategies for solving the Generalised Non-Linear Schrodinger Equation in optics by the Interaction Picture method

This report is devoted to the study and to the comparison of methods for estimation of the local error and for adaptive step-size control when solving the generalised nonlinear Schrodinger equation (GNLSE) in optics by the Interaction Picture (IP) method or by the Symmetric Split-Step (S3F) method. Namely, we propose and study the use of an embedded Runge-Kutta method to solve the nonlinear problem involved in the IP or S3F method and to deliver a local error estimate at each step of the discretisation grid for the purpose of an adaptive stepsize control. This method preserves the advantages of the RK4 method exploited in the IP or S3F methods and do not add any extra computational cost. We compare this method to other standard methods for estimating the local error such as the “step doubling method” or energy conservation based methods.