Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Athinoula A. Kosti, Simon Colreavy-Donnelly, Fabio Caraffini and Zacharias A. Anastassi
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Athinoula A. Kosti: Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Simon Colreavy-Donnelly: Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Fabio Caraffini: Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
Zacharias A. Anastassi: Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK

Mathematics, 2020, vol. 8, issue 3, 1-12

Abstract: Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

Keywords: nonlinear Schrödinger equation; periodic coefficients; varying dispersion; varying nonlinearity; Runge–Kutta pair; phase-lag; amplification error; step size control; local error estimation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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