Numerical solution of differential equations with colored noise

A closed set of coupled equations of motion for the description of time-dependent electron transport is derived. It provides the time evolution of energy-resolved quantities constructed from nonequilibrium Green functions. By means of an auxiliary-mode expansion a viable propagation scheme for finite temperatures is obtained, which allows to study arbitrary time dependences and structured reservoirs. Two illustrative examples are presented.

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“…Thereby, one has to consider another energyresolved quantity Ω(ε, ε ′ ; t) given in Eq. (22). The equations of motion are given by Eqs.…”

Section: Discussionmentioning

confidence: 99%

Propagation scheme for nonequilibrium dynamics of electron transport in nanoscale devices

Croy

Saalmann

2009

Phys. Rev. B

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“…A set of Monte Carlo simulations (MCS) is also conducted to validate the PDF method. The SDEs (2.4) are integrated numerically using a second-order strong RK scheme [21], together with an evolution equation for the Ornstein-Uhlenbeck (O-U) process that generates the exponentially correlated Gaussian fluctuations P m . The initial value of P m is drawn directly from the stationary Gaussian distribution of the O-U process.…”

Section: Computational Examplementioning

confidence: 99%

Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input

Wang

Barajas-Solano

Constantinescu³

et al. 2015

SIAM/ASA J. Uncertainty Quantification

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Wind and solar power generators are commonly described by a system of stochastic ordinary differential equations (SODEs) where random input parameters represent uncertainty in wind and solar energy. The existing methods for SODEs are mostly limited to delta-correlated random parameters, while the uncertainties from renewable generation exhibit colored noises. Here we use the probability density function (PDF) method, together with a novel large-eddy-diffusivity (LED) closure, to derive a closed-form deterministic partial differential equation (PDE) for the joint PDF of the SODEs describing a power generator with correlated-in-time power input. The proposed LED accurately captures the effect of nonzero correlation time of the power input on systems described by a divergent stochastic drift velocity. The resulting PDE is solved numerically. The accuracy of the PDF method is verified by comparison with Monte Carlo simulations.

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“…Another situation where considerable extra efficiency can be gained occurs when the SDE has only a small noise term: that is, the diffusion coefficient is small and the noise can be interpreted as a perturbation. This situation was studied by Milstein and Tretjakov (1994). Such an approximation has to focus on the drift part of the dynamics.…”

Section: Strong Taylor Approximationmentioning

confidence: 99%

“…SDEs with coloured noise were approximated by Manella and Palleschi (1989), Fox (1991) and Milstein and Tretjakov (1994).…”

Section: Further Developments and Conclusionmentioning

confidence: 99%

An introduction to numerical methods for stochastic differential equations

1999

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This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

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Numerical solution of differential equations with colored noise

Cited by 33 publications

References 16 publications

Propagation scheme for nonequilibrium dynamics of electron transport in nanoscale devices

Propagation scheme for nonequilibrium dynamics of electron transport in nanoscale devices

Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input

An introduction to numerical methods for stochastic differential equations

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