# Krylov integrators for Hamiltonian systems

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**Krylov integrators for Hamiltonian systems.** / Eirola, Timo; Koskela, Antti.

Research output: Contribution to journal › Article › Scientific › peer-review

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*BIT Numerical Mathematics*. https://doi.org/10.1007/s10543-018-0732-y

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*BIT Numerical Mathematics*. https://doi.org/10.1007/s10543-018-0732-y

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TY - JOUR

T1 - Krylov integrators for Hamiltonian systems

AU - Eirola, Timo

AU - Koskela, Antti

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider Arnoldi-like processes to obtain symplectic subspaces for Hamiltonian systems. Large dimensional systems are locally approximated by ones living in low dimensional subspaces, and we especially consider Krylov subspaces and some of their extensions. These subspaces can be utilized in two ways: by solving numerically local small dimensional systems and then mapping back to the large dimension, or by using them for the approximation of necessary functions in exponential integrators applied to large dimensional systems. In the former case one can expect an excellent energy preservation and in the latter this is so for linear systems. We consider second order exponential integrators which solve linear systems exactly and for which these two approaches are in a certain sense equivalent. We also consider the time symmetry preservation properties of the integrators. In numerical experiments these methods combined with symplectic subspaces show promising behavior also when applied to nonlinear Hamiltonian problems.

AB - We consider Arnoldi-like processes to obtain symplectic subspaces for Hamiltonian systems. Large dimensional systems are locally approximated by ones living in low dimensional subspaces, and we especially consider Krylov subspaces and some of their extensions. These subspaces can be utilized in two ways: by solving numerically local small dimensional systems and then mapping back to the large dimension, or by using them for the approximation of necessary functions in exponential integrators applied to large dimensional systems. In the former case one can expect an excellent energy preservation and in the latter this is so for linear systems. We consider second order exponential integrators which solve linear systems exactly and for which these two approaches are in a certain sense equivalent. We also consider the time symmetry preservation properties of the integrators. In numerical experiments these methods combined with symplectic subspaces show promising behavior also when applied to nonlinear Hamiltonian problems.

KW - Exponential integrators

KW - Hamiltonian Lanczos algorithm

KW - Hamiltonian systems

KW - Krylov subspace methods

KW - Symmetric integrators

KW - Symplectic integrators

UR - http://www.scopus.com/inward/record.url?scp=85055165604&partnerID=8YFLogxK

U2 - 10.1007/s10543-018-0732-y

DO - 10.1007/s10543-018-0732-y

M3 - Article

JO - BIT - Numerical Mathematics

JF - BIT - Numerical Mathematics

SN - 0006-3835

ER -

ID: 32012912