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 -