# Stochastic linear multistep methods for the simulation of chemical kinetics

## Abstract

In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach.

- Authors:

- Departamento de Informática, University of Valladolid, Valladolid (Spain)
- Department of Computer Science, University of Oxford, Oxford (United Kingdom)
- (Australia)
- School of Mathematical Sciences, Queensland University of Technology, Brisbane (Australia)

- Publication Date:

- OSTI Identifier:
- 22416091

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 142; Journal Issue: 6; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; CHEMICAL REACTION KINETICS; CHEMICAL REACTIONS; CORRELATIONS; IMPLEMENTATION; NONLINEAR PROBLEMS; STOCHASTIC PROCESSES

### Citation Formats

```
Barrio, Manuel, E-mail: mbarrio@infor.uva.es, Burrage, Kevin, School of Mathematical Sciences, Queensland University of Technology, Brisbane, and Burrage, Pamela.
```*Stochastic linear multistep methods for the simulation of chemical kinetics*. United States: N. p., 2015.
Web. doi:10.1063/1.4907008.

```
Barrio, Manuel, E-mail: mbarrio@infor.uva.es, Burrage, Kevin, School of Mathematical Sciences, Queensland University of Technology, Brisbane, & Burrage, Pamela.
```*Stochastic linear multistep methods for the simulation of chemical kinetics*. United States. doi:10.1063/1.4907008.

```
Barrio, Manuel, E-mail: mbarrio@infor.uva.es, Burrage, Kevin, School of Mathematical Sciences, Queensland University of Technology, Brisbane, and Burrage, Pamela. Sat .
"Stochastic linear multistep methods for the simulation of chemical kinetics". United States.
doi:10.1063/1.4907008.
```

```
@article{osti_22416091,
```

title = {Stochastic linear multistep methods for the simulation of chemical kinetics},

author = {Barrio, Manuel, E-mail: mbarrio@infor.uva.es and Burrage, Kevin and School of Mathematical Sciences, Queensland University of Technology, Brisbane and Burrage, Pamela},

abstractNote = {In this paper, we introduce the Stochastic Adams-Bashforth (SAB) and Stochastic Adams-Moulton (SAM) methods as an extension of the τ-leaping framework to past information. Using the Θ-trapezoidal τ-leap method of weak order two as a starting procedure, we show that the k-step SAB method with k ≥ 3 is order three in the mean and correlation, while a predictor-corrector implementation of the SAM method is weak order three in the mean but only order one in the correlation. These convergence results have been derived analytically for linear problems and successfully tested numerically for both linear and non-linear systems. A series of additional examples have been implemented in order to demonstrate the efficacy of this approach.},

doi = {10.1063/1.4907008},

journal = {Journal of Chemical Physics},

number = 6,

volume = 142,

place = {United States},

year = {Sat Feb 14 00:00:00 EST 2015},

month = {Sat Feb 14 00:00:00 EST 2015}

}

DOI: 10.1063/1.4907008

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