Stochastic discrete Hamiltonian variational integrators
File(s)
Published version
OA Location
Author(s)
Holm, DD
Tyranowski, TM
Type
Journal Article
Abstract
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.
Date Issued
2018-12-01
Date Acceptance
2018-08-07
Citation
BIT Numerical Mathematics, 2018, 58 (4), pp.1009-1048
ISSN
0006-3835
Publisher
Springer
Start Page
1009
End Page
1048
Journal / Book Title
BIT Numerical Mathematics
Volume
58
Issue
4
Copyright Statement
© 2018 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Subjects
Science & Technology
Technology
Physical Sciences
Computer Science, Software Engineering
Mathematics, Applied
Computer Science
Mathematics
Stochastic Hamiltonian systems
Variational integrators
Geometric numerical integration methods
Geometric mechanics
Stochastic differential equations
65C30
RUNGE-KUTTA METHODS
DIFFERENTIAL-EQUATIONS
ORDER CONDITIONS
QUADRATIC-INVARIANTS
SYMPLECTIC SCHEMES
SYSTEMS
math.NA
math-ph
math.DS
math.MP
math.PR
math.SG
G.1.7
0102 Applied Mathematics
0103 Numerical And Computational Mathematics
Numerical & Computational Mathematics
Publication Status
Published
Date Publish Online
2018-08-16