New rational methods for the numerical solution of first order initial value problem
Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form Y¹= ʎy,y(a)=ɳ, ʎ є C , Rc(λ)<0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been...
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| Main Authors: | , , |
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| Format: | Monograph |
| Language: | English |
| Published: |
Universiti Utara Malaysia
2012
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| Subjects: | |
| Online Access: | http://repo.uum.edu.my/24777/1/12412.pdf http://repo.uum.edu.my/24777/ |
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| Summary: | Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form
Y¹= ʎy,y(a)=ɳ, ʎ є C , Rc(λ)<0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially fitted.Recently, a class of one-step exponential-rational methods (ERMs) were discovered.Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability.Several numerical experiments showed that ERMs is more accurate than existing rational methods in solving general initial value problem. However, ERMs have several weaknesses: i) every ERM is non-uniquely defined; ii) may return
complex values; and iii) less accurate numerical solution when solving problem whose solution possesses singularity.Therefore, the first purpose of this study is to modify the original ERMs so that the first two weaknesses will be overcomed.
Theoretical analyses such as consistency, stability and convergence of the modified
ERMs are presented.Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.The second purpose of this study is to overcome the third weakness of the original ERMs where a variable step-size strategy is proposed to improve the accuracy ERMs.The procedures of the strategy are detailed out in this report.Numerical experiments have revealed that the affects from the implementation of the strategy is less obvious. |
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