A method of Lie-symmetry GL(n ,RR) for solving non-linear dynamical systems

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)SOo(n,1). Here we derive a more fundamental GL(n,R)GL(n,R) dynamics for xx, and develop a relevant Lie-group scheme based on the Lie-symmetry GL(n,R)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)GL(n,R) is equivalent to the GPS. Moreover, when one uses an implicit scheme based on GL(n,R)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)GL(n,R) and SO(n) Lie-group schemes have superior efficiency, accuracy and stability.

► The Lie-symmetry GL(n,R)GL(n,R) can cope all non-linear dynamical systems.
► The non-linear dynamical system was solved by GL(n,R)GL(n,R) integration method.
► SO(n)SO(n) integration method was used on rotational vector field.
► The present implicit method is convergent very fast, effective and accurate.