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

Abstract

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field methods to preserve multiple invariants simultaneously. Based on the prior estimate for high order moments of the modification coefficient, the mean square convergence order 1 of proposed methods is proved in the case of commutative noises. In addition, the effect of quadrature formula on the mean square convergence order and the preservation of invariants for the modified averaged vector field methods is considered. Numerical experiments are performed to verify the theoretical analyses and to show the superiority of the proposed methods in long time simulation.

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems With Holonomic Constraints

“…The theory of symplectic integrators for Hamiltonian systems has become more complete with the establishment of backward error analysis [4,28] and discrete KAM theory [25]. Recently, some related numerical methods such as multi-symplectic methods for Hamiltonian partial differential equations [23] and stochastic symplectic methods for stochastic Hamiltonian systems [10] have been well developed. In addition to the symplectic property, another extremely important feature of the Hamiltonian system is the conservation of energy.…”

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

“…They have a number of advantages and provide reliable numerical solutions. Therefore, GNI for SDEs with geometric attributes attracted a considerable attention [4,14,19,24,33]. In this work, we are mainly interested in SDEs with the conserved energy and momentum.…”

“…On the other hand, to the best of our knowledge there are only a few researches concerning conservative numerical methods for SDEs with multiple conserved quantities. Thus Chen et al [4] modified the stochastic averaging vector field (AVF) method to preserve multiple conserved quantities, Zhou et al [34] introduced multiple Lagrange multipliers and directly applied a linear projection method to SDEs with multiple conserved quantities. In comparison to the case of a single conserved quantity, this can increase the computational costs.…”

Drift-preserving numerical integrators for stochastic Hamiltonian systems

“…Indeed, many publications on energy-preserving numerical integration of stochastic (canonical) Hamiltonian systems have lately appeared. Without being too exhaustive, we mention [7,11,17,26] as well as the works [25,40,8] on invariant preserving schemes. Observe, that such extensions describe Hamiltonian motions perturbed by a multiplicative white noise in the sense of Stratonovich.…”