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

Hong, Jialin; Xu, Dongsheng; Wang, Peng

doi:10.1016/j.apnum.2014.08.003

Cited by 33 publications

(25 citation statements)

References 19 publications

(24 reference statements)

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“…Remark 3.2. The conditions for the first mean-square order SRK methods (conditions 1-4 in Theorem 3.1) agree with the results in [21,18] when considering the additive noise case.…”

Section: Order Conditions Of Srk Methods For Sdes With Additive Noisesupporting

confidence: 77%

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Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Zhou

Zhang

Hong

et al. 2017

Journal of Computational and Applied Mathematics

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In this paper, we construct stochastic symplectic Runge-Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.

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“…Remark 3.2. The conditions for the first mean-square order SRK methods (conditions 1-4 in Theorem 3.1) agree with the results in [21,18] when considering the additive noise case.…”

Section: Order Conditions Of Srk Methods For Sdes With Additive Noisesupporting

confidence: 77%

“…Remark 4.2. Symplectic conditions for Hamiltonian systems with multiplicative noise can be found in [18,21]. However, more conditions similar to (4.1) must be added.…”

Section: Symplectic Conditions Of Srk Methods For Stochasticmentioning

confidence: 99%

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Zhou

Zhang

Hong

et al. 2017

Journal of Computational and Applied Mathematics

Self Cite

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show abstract

“…Example 4. A simple example for an SPRK method fulfilling (23) is the stochastic Störmer-Verlet method [14], given by the following tableau: .…”

Section: Example 3 (Synchrotron Oscillations)mentioning

confidence: 99%

General order conditions for stochastic partitioned Runge–Kutta methods

2017

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Abstract. In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions.

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“…which satisfy X k t = X k 0 . As an alternative to Scheme 1, we can solve (16) by schemes preserving quadratic invariants (see, e.g., [1,5,29] and [16,19,32]). Since −i L j (t) is a symmetric operator, according to (16) we have…”

Section: The Euler-exponential Schemementioning

confidence: 99%

Numerical solution of stochastic quantum master equations using stochastic interacting wave functions

Mora¹,

Fernández

Biscay

2018

Journal of Computational Physics

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We develop a new approach for solving stochastic quantum master equations with mixed initial states. First, we obtain that the solution of the jump-diffusion stochastic master equation is represented by a mixture of pure states satisfying a system of stochastic differential equations of Schrödinger type. Then, we design three exponential schemes for these coupled stochastic Schrödinger equations, which are driven by Brownian motions and jump processes. Hence, we have constructed efficient numerical methods for the stochastic master equations based on quantum trajectories. The good performance of the new numerical integrators is illustrated by simulations of two quantum measurement processes. t 0 Tr R m (s) * R m (s) ρ s− ds (see , e.g., [12, 31, 46]) such that N 1 , . . . , N M have no common jumps, and

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

Cited by 33 publications

References 19 publications

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

General order conditions for stochastic partitioned Runge–Kutta methods

Numerical solution of stochastic quantum master equations using stochastic interacting wave functions

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