Abstract
This paper reviews the Lie Group’s usage and applications in numerical methods for ordinary differential equations. We present comparisons between the midpoint symplectic and the usual Runge–Kutta methods for distinct dynamical behaviors ranging from integrable to chaotic regimes. Simulation results show that the first has better precision in the regular region of the phase space, according to a statistical indicator defined in this work. Close to the homoclinic crossing, this performance degrades sharply.
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