English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The paper investigates the exponential discrete gradient methods simulating stochastic Hamiltonian systems with invariant Hamiltonians. We show that the methods are nearly symplectic, and nearly preserve the invariant Hamiltonians. Numerical experiments illustrate effectiveness and efficiency of the simulation methods. Introduction Hamiltonian systems are very important dynamical systems. All physical processes where the dissipation can be neglected can be described as Hamiltonian systems ([1]). Stochastic Hamiltonian systems (SHSs) are Hamiltonian systems with certain random disturbances, and are described as ([2]) ∑ ∘ , 0 , (1) where ∈ , 0 0 , 1,⋯ , : → are real functions ensuring existence and uniqueness of the solution of (1), called the Hamiltonians of the system (1). , ⋯ , is an m-dimensional standard Wiener process. As their deterministic counterpart, the stochastic Hamiltonian systems have an intrinsic structure, the symplectic structure, depicted as ([1,2,3]) , ∀ 0. (2) Geometrically, (2) means area preservation in the phase space along the phase flow of the Hamiltonian systems. Exact solutions of Hamiltonian systems are in general difficult to be found, therefore numerical simulations become an import tool of investigating such systems. Numerous numerical methods were developed for deterministic and stochastic Hamiltonian systems, in particular, people pursue symplectic methods for such systems (see e.g. [1,2,3,4] and references therein), methods that can preserve the symplectic structure (2) of the system, characterized by , ∀ 0. (3) More general, methods that can preserve structures of the original systems are called structure-preserving algorithms, such as the symplectic methods for Hamiltonian systems, energy-preserving methods for systems with invariant energy, and so on ([3]). Such methods turn out to be superior than other general-purpose methods, in qualitative behavior and long-time simulation. In this paper, we investigate the stochastic exponential discrete gradient (SEDG) methods applied to the stochastic Hamiltonian systems with invariant Hamiltonians. We examine their properties in preserving the symplecticity and the Hamiltonians, and show that they can nearly preserve the symplecticity and the Hamiltonians within error of root mean-square order 1. We also illustrate

Exponential Discrete Gradient Schemes for Simulating Stochastic Hamiltonian Systems Preserving Hamiltonian Functions