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Article information
2015 , Volume 20, ¹ 3, p.46-57
Paasonen V.I., Fedoruk M.P.
Three-level non-iterative high accuracy scheme for Ginzburg - Landau equation
This paper presents a generalization of compact difference scheme previously developed and investigated by the authors for one-dimensional nonlinear Schrödinger equation to the case of the Ginzburg -Landau equation. We apply the increased order of accuracy as a very useful tool for improvement of the quality of calculations. The scheme approximates the Ginzburg-Landau equation with the second-order for the evolutionary variable and with the fourth order for the "slow time". The scheme is essentially of a two-level, but the scheme uses a double step at three levels, with the non-linear approximation on the right side at the middle level. This approach do not require the need to iterate on the nonlinearity at all steps, except the first, and thus saves computing resources. To compute the solution in the first step we proposed to use a two-level iterative scheme providing the same order of approximation as for the main scheme. Stability of difference schemes was examined in the linear approximation using the analysis of variance of the behavior harmonics. The paper presents the results of calculations on the sequence of the test problems which use the condensing grids with known exact solutions obtained earlier by Ahmediev and Afanasiev in their famous work. A comparison with the results obtained by the three-layered modified Crank-Nicolson scheme with approximation of the second order. The above graphic and tabular material evidence of significant advantages of a compact difference scheme.
[full text]
Keywords: Shrodinger equation, compact difference scheme, high-order accuracy scheme, nonlinear fiber optics, fiber laser, Ginzburg - Landau equation
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. Three-level non-iterative high accuracy scheme for Ginzburg - Landau equation // Computational technologies. 2015. V. 20. ¹ 3. P. 46-57
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