Energy conservative stochastic difference scheme for stochastic Hamilton dynamical systems

Abstract

An energy conservative stochastic difference scheme is proposed for a one-dimensional stochastic Hamilton dynamical system governed by a stochastic differential equations in which the energy function, i.e. Hamiltonian, becomes a conserved quantity. The scheme is given by an stochastic extension of Greenspan's scheme which leaves Hamiltonians numerically invariant for deterministic Hamilton dynamical systems. The local error of accuracy of numerical solutions derived from the stochastic difference scheme is investigated. An illustrative numerical experiment for the scheme to a simple stochastic Hamilton system is also given.

Projection methods for stochastic differential equations with conserved quantities

“…Here are some examples of stochastic systems with conserved quantity. One is the stochastic canonical Hamiltonian system (d = 2n) [18]…”

“…In this aspect, a range of numerical methods have been proposed to preserve different properties of SDEs with special forms (e.g. [9,15,18]). Among these properties is the conserved quantity which is intrinsic and essential for some stochastic systems.…”

Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations

“…For conservative SDEs with one invariant, there have been many works related to numerical methods in recent years. On the one hand, aiming at the SDEs with single noise, [12] proposes an energy-preserving difference method for stochastic Hamiltonian systems and analyzes the local errors. Based on the equivalent skew gradient (SG) form for conservative SDEs with one invariant, [6] proposes direct discrete gradient methods and indirect discrete gradient methods, and proves that these two kinds of methods are of mean square order 1.…”

High-Order Energy-Preserving Methods for Stochastic Poisson Systems

“…[1,4,5,7,10,18,[23][24][25][26]. Nevertheless, energy-preserving methods for stochastic systems are less developed, although for stochastic ordinary differential equations one can note difference methods [21], discrete gradient methods [11,15], projection methods [15,27], averaged vector field methods [6] and a few others.…”