A method of Lie-symmetry GL(n,[formula omitted]) for solving non-linear dynamical systems

Abstract

The group-preserving scheme (GPS) developed in Liu [1] for solving a non-linear dynamical system x˙=f(x,t) was based on the Lie-symmetry SOo(n,1). Here we derive a more fundamental GL(n,R) dynamics for x, and develop a relevant Lie-group scheme based on the Lie-symmetry GL(n,R), which is a Lie-group set large enough to cope all non-linear ordinary differential equations (ODEs). We find that the first-order explicit scheme based on GL(n,R) is equivalent to the GPS. Moreover, when one uses an implicit scheme based on GL(n,R), it converges very fast at each time marching step and the accuracy is raised several orders than the explicit method. For the dynamical system endowed with a rotational vector field we also develop an implicit SO(n) Lie-group integration method. Several numerical examples are examined, showing that the GL(n,R) and SO(n) Lie-group schemes have superior efficiency, accuracy and stability.