Krylov integrators for Hamiltonian systems

TL;DR

This paper develops Krylov-based symplectic integrators for Hamiltonian systems, enabling efficient local approximations and energy-preserving numerical solutions, with promising results on nonlinear problems.

Contribution

It introduces Krylov subspace methods tailored for Hamiltonian systems, demonstrating their effectiveness in energy preservation and approximation in both linear and nonlinear cases.

Findings

Excellent energy preservation in local small-dimensional systems

Equivalent approaches for linear systems using Krylov subspaces

Promising behavior observed in nonlinear Hamiltonian problems

Abstract

We consider Arnoldi like processes to obtain symplectic subspaces for Hamiltonian systems. Large systems are locally approximated by ones living in low dimensional subspaces; we especially consider Krylov subspaces and some extensions. This will be utilized in two ways: solve numerically local small dimensional systems or in a given numerical, e.g. exponential, integrator, use the subspace for approximations of necessary functions. In the former case one can expect an excellent energy preservation. For the latter this is so for linear systems. For some second order exponential integrators we consider these two approaches are shown to be equivalent. In numerical experiments with nonlinear Hamiltonian problems their behaviour seems promising.