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

Athinoula A. Kosti Orcid Logo, Simon Colreavy-Donnelly Orcid Logo, Fabio Caraffini Orcid Logo, Zacharias A. Anastassi Orcid Logo

Mathematics, Volume: 8, Issue: 3, Start page: 374

Swansea University Author: Fabio Caraffini Orcid Logo

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DOI (Published version): 10.3390/math8030374

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...

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Published in: Mathematics
ISSN: 2227-7390
Published: MDPI AG 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa60956
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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
College: College of Science
Funders: This research received no external funding.
Issue: 3
Start Page: 374