Projected Runge-Kutta methods for differential equations on matrix Lie groups

For differential equations on a matrix Lie group it is known that most traditional methods do not keep the numerical solutions in the Lie group. To obtain numerical solutions with the correct qualitative behaviour we solve overdetermined systems of differential-algebraic equations, including differential equations on the corresponding Lie algebra, by standard Runge-Kutta methods with projection in the Lie algebra. Theoretical analysis and numerical experiment demonstrate that this new class of Lie-group invariant methods is more efficient than previous ones