Vasilyev E.I., Vasilyeva T.A., Kiseleva M.N. L-stability of multi-implicit methods of 8-th order for differential stiff systems

http://dx.doi.org/10.15688/jvolsu1.2013.1.6

Vasilyev Evgeniy Ivanovich

Doctor of Physics and Mathematics, Professor, Department of Computer Science and Optimal Control, Volgograd State University
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Prospect Universitetskij, 100, 400062 Volgograd, Russian Federation

Vasilyeva Tatiana Anatolievna

Associate Professor, Department of Computer Science and Optimal Control,
Volgograd State University
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Prospect Universitetskij, 100, 400062 Volgograd, Russian Federation


Kiseleva Maria Nikolaevna
Postgraduate student, Department of Computer Science and Optimal Control,
Volgograd State University
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Prospect Universitetskij, 100, 400062 Volgograd, Russian Federation

Abstract. The new set of absolutely stable difference schemes for a numerical
solution of ODEs stiff systems (1) is submitted.

201318 AppliedMath pic1

The main feature of the set is the multi-implicit finite differences with the second de-
rivatives of the desired solution. The expanded two-parameter (α, β) set of 3ISD-schemes (2–3) is studied in more details in this paper.
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At arbitrary (α, β) parameters last difference equation in (2) has 8-th order of ap-
proximating.
We found that the set of absolutely stable 3ISD-schemes includes two one parameter families: the set of the L-stable schemes and the set of the schemes of heightened accuracy for linear problems. For example:

at values α=1/540, β=1/1080we have A-stable scheme with 10-th order of approximating, 

at values α=1/54, β=-1/135we have L1-stable scheme with 9-th order of approximating, 

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L-stability of multi-implicit methods of 8-th order for differential stiff systems by Vasilyev E.I., Vasilyeva T.A., Kiseleva M.N. is licensed under a Creative Commons Attribution 4.0 International License.

at values α=1/54, β=-1/216we have L2-stable scheme with 8-th order of approximating.
The testing of this difference schemes on linear and non-linear problems with a different stiff power is conducted. The errors of a numerical solution as functions of integration step size are computed in numerical experiments. These results demonstrate high quality of stability and accuracy of the suggested 3ISD-schemes.

Key words: L-stability, А-stability, stiff systems, implicit methods, multiimplicit methods, methods with second derivative.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (18) 2013 pp. 70-83

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