Numerical simulation of the non-linear Schrödinger equation A comparative study based on the summation-by-parts method in space and time

Abstract: In this work, we present the numerical implementation of various time marching methods based on the summation-by-parts (SBP) method for the time integration of nonlinear dispersive systems. More specifically, we consider the nonlinear Schrödinger (NLS) equation that accepts soliton or soliton-like solution as a primary case study. The spatial discretization is performed via the SBP method with the projection technique to enforce the boundary conditions strongly. We show that the method in combination with the projection technique represents a robust, accurate, and highly stable numerical method for the treatment of Nonlinear Schrödinger-type systems. The imposition of the initial data for the SBP-based time marching methods can, equivalently to SBP-spatial discretization, be combined with either the simultaneous approximation term (SAT) or the projection technique. In contrast to using the SAT for the imposition of the initial data, the projection technique represents a part of recent progress in the field and has not been studied in the literature. To the best of our knowledge, no investigation of the numerical efficiency of the SBP-based time marching methods has been conducted yet in the context of the nonlinear initial boundary value problem (IBVP). Therefore, we present in this work a thorough numerical efficiency analysis for the NLS-type systems using both the SAT and the projection technique with the classical fourth-order Runge-Kutta as a standard of comparison.