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We present a multiscale integrator for Hamiltonian systems with slowlyvarying quadratic stiff potentials that uses coarse timesteps (analogous towhat the impulse method uses for constant quadratic stiff potentials). Thismethod is based on the highly-non-trivial introduction of two efficientsymplectic schemes for exponentiations of matrices that only require O(n)matrix multiplications operations at each coarse time step for a preset smallnumber n. The proposed integrator is shown to be (i) uniformly convergent onpositions; (ii) symplectic in both slow and fast variables; (iii) well adaptedto high dimensional systems. Our framework also provides a general method foriteratively exponentiating a slowly varying sequence of (possibly highdimensional) matrices in an efficient way.

From efficient symplectic exponentiation of matrices to symplectic  integration of high-dimensional Hamiltonian systems with slowly varying  quadratic stiff potentials