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

TL;DR

This paper introduces a multiscale symplectic integrator for high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials, leveraging efficient matrix exponentiation schemes to achieve uniform convergence and symplecticity.

Contribution

It develops a novel multiscale integrator that efficiently handles high-dimensional systems with slowly varying potentials using new symplectic matrix exponentiation methods.

Findings

Achieves uniform convergence on positions.

Maintains symplectic structure in both variables.

Efficiently handles high-dimensional systems.

Abstract

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