Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric Integrators that respect these symmetries. Extending thesemethods to Euler and Navier-Stokes is an outstanding research problem in fluid mechanics. Therefore, a short review of geometric integrators for ODEs is given in this last chapter. Exponential coordinates on a Lie group are explained; the formula for differentiating a matrix exponential is given and used to derive the first few terms of the Magnus expansion. Geometric integrators corresponding to the Euler and trapezoidal methods for ODEs are given.
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