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Nonstandard Group-Preserving Schemes for Very Stiff Ordinary Differential Equations

Chein-Shan Liu1
Department of Mechanical and Mechatronic Engineering, Taiwan Ocean University, Keelung, Taiwan, E-mail:

Computer Modeling in Engineering & Sciences 2005, 9(3), 255-272.


The group-preserving scheme developed by Liu (2001) for calculating the solutions of k-dimensional differential equations system adopted the Cayley transform to formulate the Lie group from its Lie algebra A ∈ so(k,1). In this paper we consider a more effective exponential mapping to derive exp(hA). In order to overcome the difficulty of numerical instabilities encountered by employing group-preserving schemes on stiff differential equations, we further combine the nonstandard finite difference method into the group-preserving schemes to obtain unconditional stable numerical methods. They provide single-step explicit time integrators for stiff differential equations. Several numerical examples are examined, some of which are compared with exact solutions showing that the nonstandard group-preserving schemes have good computational efficiency and certain accuracy.


Stiff differential equations, nonstandard group-preserving scheme, A-stable, L-stable

Cite This Article

Liu, C. (2005). Nonstandard Group-Preserving Schemes for Very Stiff Ordinary Differential Equations. CMES-Computer Modeling in Engineering & Sciences, 9(3), 255–272.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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