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

Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK
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Mathematics 2020, 8(3), 374; https://doi.org/10.3390/math8030374 (registering DOI)
Received: 11 February 2020 / Revised: 1 March 2020 / Accepted: 4 March 2020 / Published: 7 March 2020
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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. View Full-Text
Keywords: nonlinear Schrödinger equation; periodic coefficients; varying dispersion; varying nonlinearity; Runge–Kutta pair; phase-lag; amplification error; step size control; local error estimation nonlinear Schrödinger equation; periodic coefficients; varying dispersion; varying nonlinearity; Runge–Kutta pair; phase-lag; amplification error; step size control; local error estimation
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Kosti, A.A.; Colreavy-Donnelly, S.; Caraffini, F.; Anastassi, Z.A. Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients. Mathematics 2020, 8, 374.

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