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Runge-Kutta methods on Lie groups. (English) Zbl 0904.65077

The author constructs generalized Runge-Kutta (RK) methods for solving differential equations evolving on a Lie group. Since the methods use intrinsic operations on the group, the numerical solutions evolve on the correct manifold, what, in general, is not true in the case of embedded methods.
In order to achieve a given order the presented family of RK methods satisfied two criteria: coefficients \(a_{ij}\) and \(b_j\) satisfy the classical order conditions, and it is necessary to construct functions which correct certain non-commutative effects to the given order. After excellent mathematical background the author presents explicitly 3rd and 4th order algorithms and a numerical example.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
22E99 Lie groups
34A34 Nonlinear ordinary differential equations and systems
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