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On Crank – Nicolson schemes for non-stationary problems with operators reducible to skew-symmetric form

On Crank – Nicolson schemes for non-stationary problems with operators reducible to... - Higher-order accurate Crank-Nicolson schemes are obtained for solving homogeneous and inhomogeneous Cauch/s problems with operators reducible to a skew-symmetric form. The phase error is diminished owing to a special algorithm of choosing time-steps. For homogeneous problems, by using complex time-steps we obtain difference schemes with an accuracy of ( ), ( ). On the basis of the difference schemes the algorithms for numerical solution of diffraction problems are presented and the results of calculations are given. The Crank-Nicolson difference schemes are known to retain the solution norm in the problems with skew-symmetric operators (see, for example, [4]). Therefore an error arising in calculations with the use of these schemes is of a phase nature. In this paper, owing to a special choice of the time-step sequence , we construct Crank-Nicolson difference schemes with an accuracy of ^(4), <^(6), <^(8) in the phase, where is the average size of the time-step. Some of the schemes considered are schemes with complex steps. Difference schemes are suggested for solving a diffraction problem. 1. INCREASING THE ACCURACY OF CRANK-NICOLSON DIFFERENCE SCHEMES BY CHANGING VARIABLE TIME-STEPS Let A be an operator reducible to a skew-symmetric form with the aid of an invertible http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Russian Journal of Numerical Analysis and Mathematical Modelling de Gruyter

On Crank – Nicolson schemes for non-stationary problems with operators reducible to skew-symmetric form

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References (3)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0927-6467
eISSN
1569-3988
DOI
10.1515/rnam.1993.8.1.47
Publisher site
See Article on Publisher Site

Abstract

- Higher-order accurate Crank-Nicolson schemes are obtained for solving homogeneous and inhomogeneous Cauch/s problems with operators reducible to a skew-symmetric form. The phase error is diminished owing to a special algorithm of choosing time-steps. For homogeneous problems, by using complex time-steps we obtain difference schemes with an accuracy of ( ), ( ). On the basis of the difference schemes the algorithms for numerical solution of diffraction problems are presented and the results of calculations are given. The Crank-Nicolson difference schemes are known to retain the solution norm in the problems with skew-symmetric operators (see, for example, [4]). Therefore an error arising in calculations with the use of these schemes is of a phase nature. In this paper, owing to a special choice of the time-step sequence , we construct Crank-Nicolson difference schemes with an accuracy of ^(4), <^(6), <^(8) in the phase, where is the average size of the time-step. Some of the schemes considered are schemes with complex steps. Difference schemes are suggested for solving a diffraction problem. 1. INCREASING THE ACCURACY OF CRANK-NICOLSON DIFFERENCE SCHEMES BY CHANGING VARIABLE TIME-STEPS Let A be an operator reducible to a skew-symmetric form with the aid of an invertible

Journal

Russian Journal of Numerical Analysis and Mathematical Modellingde Gruyter

Published: Jan 1, 1993

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