Approximating the exponential from a Lie algebra to a Lie group

Consider a differential equation $y^{'}=A(t,y)y, y(0)=y_0$ with $y_ 0\in \mathrm{G}$and $A:\mathbb{R}^{+}\times \mathrm{G}\rightarrow \mathfrak{g}$, where $\mathfrak{g}$ is a Lie algebra of the matricial Lie group $\mathrm{G}$. Every $B\in \mathfrak{g}$ can be mapped to $\mathrm{G}$ by the matrix exponential map $\operatorname{exp}{(tB)}$ with $t\in \mathbb{R}$.

Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation $y_n$ of the exact solution $y (t_n)$, $t_n \in \mathbb{R}^{+}$, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value $y_0$. This ensures that $y_n\in \mathrm{G}$.

When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of $\operatorname{exp}{(tB)}$ plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby $\operatorname{exp}{(tB)}$ is approximated by a product of simpler exponentials.

In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of $\mathfrak{g}$ and $\mathrm{G}$ are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.