Optimized Lie-group methods for differential equations

Fernando Casas

Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain

casas@mat.uji.es

Contributed talk

In this talk we present numerical algorithms for solving the initial value problem

$$Y' = A(t,Y) Y, \qquad Y(t_0) = Y_0 \in \mathcal{G}$$ (1)

with $A: [t_0, \infty[ \times \mathcal{G} \longrightarrow \mathfrak{g}$. Here $\mathcal{G}$ is a matrix Lie group and $\mathfrak{g}$ stands for the associated Lie algebra. Such methods provide numerical approximations that evolve also on $\mathcal{G}$ and are optimal with respect to the number of commutators required. This reduction in the computational cost of the algorithms is achieved by developing a general optimization technique in a graded free Lie algebra. Both linear and nonlinear differential equations are considered.

In the linear case we obtain a 6th-order scheme based on the Magnus expansion which requires only three commutators and one matrix exponential per time step. The method is generalized to nonhomogeneous equations with the minimum computational effort.

The optimization technique can also be applied to the general (nonlinear) case (1). In particular, a 5th-order Runge-Kutta-Munthe-Kaas method involving the minimum number of commutators is presented.

Finally, other generalizations of the Magnus expansion for particular nonlinear equations are also discussed.