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Article information
2022 , Volume 27, ¹ 6, p.58-69
Paasonen V.I., Fedoruk M.P.
The compact iteration-free difference scheme for the two-dimensional Ginzburg - Landau equation
This paper represents an extension of the series of articles which addressed high-accuracy compact schemes for the equations of nonlinear wave optics. The subject of the study is both the Dirichlet problem and the periodic boundary value problem for the Ginzburg – Landau equation in a two-dimensional rectangular domain. For simplicity, we consider an equation with only third- degree nonlinearity, although the absence of terms of the fifth degree is not a limiting factor for the application of the proposed algorithm in the general case. For the numerical solution of the problem, a compact fourth-order difference scheme similar to the classical Mikeladze scheme for the two-dimensional heat conduction equation is constructed. The nonlinear term problem is solved without an iterative process. Instead, the value of the solution on the upper layer is pre-computed from the explicit scheme, and the obtained result is substituted in the right side of the main scheme to approximate the nonlinear term. Thus, the scheme as a whole has a predictor-corrector structure. By the way, a similar non-intermediate version of the Crank-Nicholson scheme is formulated. By means of harmonic analysis, the stability of both schemes in linear approximation is investigated. The results of calculations using the compact scheme are presented in comparison with calculations using the Crank – Nicholson scheme. Problems with three different initial amplitude and initial phase assignments, as well as the problem with the exact plane wave solution, were taken as test model problems. By calculations on a sequence of condensing grids, posterior estimates of solution errors and real orders of accuracy in uniform and quadratic norms are obtained.
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Keywords: Ginzburg - Landau equation, compact difference scheme, high-order accuracy, iteration-free scheme, nonlinear fiber optics
doi: 10.25743/ICT.2022.27.6.006
Author(s):
Paasonen Viktor Ivanovich
PhD. , Associate Professor
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: paas@ict.nsc.ru
Fedoruk Mikhail Petrovich
Dr. , Academician RAS, Professor
Position: Chancellor
Office: Novosibirsk State University, Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, str. Pirogova, 2
Phone Office: (3832) 349105
E-mail: mife@net.ict.nsc.ru
SPIN-code: 4929-8753 References:
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Bibliography link:
Paasonen V.I., Fedoruk M.P. The compact iteration-free difference scheme for the two-dimensional Ginzburg - Landau equation // Computational technologies. 2022. V. 27. ¹ 6. P. 58-69
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