The fourth-order group preserving methods for the integrations of ordinary differential equations

The group-preserving schemes developed by Liu (2001) for integrat- ing ordinary differential equations system were adopted the Cayley transform and Padé approximants to formulate the Lie group from its Lie algebra. However, the accuracy of those schemes is not better than second-order. In order to increase the accuracy by employing the group-preserving schemes on ordinary differen- tial equations, according to an efficient technique developed by Runge and Kutta to raise the order of accuracy from the Euler method, we combine the Runge- Kutta method on the group-preserving schemes to obtain the higher-order numeri- cal methods of group-preserving type. They provide single-step explicit time inte- grators for differential equations. Several numerical examples are examined, show- ing that the higher-order group-preserving schemes have good computational effi- ciency and high accuracy. Copyright © 2009 Tech Science Press.