An Embedded Split-Step method for solving the nonlinear Schrodinger equation in optics
Résumé
In optics the nonlinear Schrodinger equation (NLSE) which modelize wave propagation in an optical fiber is the most widely solved by the Symmetric Split-Step method. The practical efficiency of the Symmetric Split-Step method is highly dependent on the computational grid points distribution along the fiber, therefore an efficient adaptive step-size control strategy is mandatory. The most common approach for step-size control is the ''step-doubling'' approach. It provides an estimation of the local error at each computational grid point in order to set the next grid point in the best way to match a user predefined tolerance. The step-doubling approach increases of around 50 % the computational cost of the Symmetric Split-Step method. Alternatively there exists in optics literature other approaches based on the observation along the propagation length of the behavior of a given optical quantity. The step-size at each computational step is set so as to guarantee that the known properties of the quantity are preserved. These approaches derived under specific physical assumptions are low cost but suffer from a lack of generality. In this paper we present a new method for estimating the local error in the Symmetric Split-Step method when solving the NLSE. It conciliates the advantages of the step-doubling approach in term of generality and rigor but without the drawback of requiring a significant extra computational cost. The method is related to Embedded Split-Step methods for nonlinear evolution problems.
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