Energy-preserving integrators for stochastic Poisson systems

Abstract

A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [E.

Preservation of quadratic invariants of stochastic differential equations via Runge–Kutta methods

“…There are many references concerning this problem. [5] studies the midpoint (trapezoidal) methods preserving the first and second moments for linear SDEs; [17] proposes the conserving energy difference scheme for stochastic dynamical systems; [16] constructs symplectic numerical schemes for stochastic canonical Hamiltonian problems; [14,15] develop the generating functions for stochastic symplectic methods; [12] investigates the boundary preserving semianalytic numerical algorithms for SDEs; [7] designs invariants-preserving methods for SDEs by using the discrete gradient approach; The recent works [3] and [2] propose a new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix and present a novel conservative method for numerical computation of general stochastic differential equations in the Stratonovich sense with a conserved quantity, respectively.…”

Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

“…where c is a real-valued parameter. Note that it is also called Kubo oscillator and is a typical stochastic Hamiltonian system with multiplicative noise [20,21]. It is easy to check that (5.2) has a quadratic conserved quantity…”

Projection methods for stochastic differential equations with conserved quantities

“…Applying the discrete gradient method to (35), we get where∇I(x, X) is some kind of discrete gradient. This method has mean-square order 1, and for details refer to [3,5,11]. Next we consider the projection method with Euler-Maruyama method X as its supporting method and choose ∇I(x) as the projection direction:…”