Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise

In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM (s,r) methods to construct symplectic Runge–Kutta methods for all values of s and r with s≥r. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2012-10, Vol.236 (16), p.3920-3930
Main Authors: Burrage, Kevin, Burrage, Pamela M.
Format: Article
Language:English
Subjects:
Additives
Computation
Computer simulation
Construction
Mathematical models
Noise
Runge-Kutta method
Runge–Kutta methods
Stochastic Hamiltonian problems
Stochasticity
Symplecticity
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Summary:In this paper we extend the ideas of Brugnano, Iavernaro and Trigiante in their development of HBVM (s,r) methods to construct symplectic Runge–Kutta methods for all values of s and r with s≥r. However, these methods do not see the dramatic performance improvement that HBVMs can attain. Nevertheless, in the case of additive stochastic Hamiltonian problems an extension of these ideas, which requires the simulation of an independent Wiener process at each stage of a Runge–Kutta method, leads to methods that have very favourable properties. These ideas are illustrated by some simple numerical tests for the modified midpoint rule.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2012.03.007