The Fourth-Order Group Preserving Methods for the Integrations of Ordinary Differential Equations

Abstract

The group-preserving schemes developed by Liu (2001) for integrat- ing ordinary differential equations system were adopted the Cayley transform and Pad´ e 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.