# Hybrid discrete/continuum algorithms for stochastic reaction networks

## Abstract

Direct solutions of the Chemical Master Equation (CME) governing Stochastic Reaction Networks (SRNs) are generally prohibitively expensive due to excessive numbers of possible discrete states in such systems. To enhance computational efficiency we develop a hybrid approach where the evolution of states with low molecule counts is treated with the discrete CME model while that of states with large molecule counts is modeled by the continuum Fokker–Planck equation. The Fokker–Planck equation is discretized using a 2nd order finite volume approach with appropriate treatment of flux components. The numerical construction at the interface between the discrete and continuum regions implements the transfer of probability reaction by reaction according to the stoichiometry of the system. The performance of this novel hybrid approach is explored for a two-species circadian model with computational efficiency gains of about one order of magnitude.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22382165

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 281; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; EFFICIENCY; FOKKER-PLANCK EQUATION; GAIN; INTERFACES; MATHEMATICAL MODELS; MATHEMATICAL SOLUTIONS; MOLECULES; PERFORMANCE; PROBABILITY; STOCHASTIC PROCESSES; STOICHIOMETRY

### Citation Formats

```
Safta, Cosmin, E-mail: csafta@sandia.gov, Sargsyan, Khachik, E-mail: ksargsy@sandia.gov, Debusschere, Bert, E-mail: bjdebus@sandia.gov, and Najm, Habib N., E-mail: hnnajm@sandia.gov.
```*Hybrid discrete/continuum algorithms for stochastic reaction networks*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.10.026.

```
Safta, Cosmin, E-mail: csafta@sandia.gov, Sargsyan, Khachik, E-mail: ksargsy@sandia.gov, Debusschere, Bert, E-mail: bjdebus@sandia.gov, & Najm, Habib N., E-mail: hnnajm@sandia.gov.
```*Hybrid discrete/continuum algorithms for stochastic reaction networks*. United States. doi:10.1016/J.JCP.2014.10.026.

```
Safta, Cosmin, E-mail: csafta@sandia.gov, Sargsyan, Khachik, E-mail: ksargsy@sandia.gov, Debusschere, Bert, E-mail: bjdebus@sandia.gov, and Najm, Habib N., E-mail: hnnajm@sandia.gov. Thu .
"Hybrid discrete/continuum algorithms for stochastic reaction networks". United States.
doi:10.1016/J.JCP.2014.10.026.
```

```
@article{osti_22382165,
```

title = {Hybrid discrete/continuum algorithms for stochastic reaction networks},

author = {Safta, Cosmin, E-mail: csafta@sandia.gov and Sargsyan, Khachik, E-mail: ksargsy@sandia.gov and Debusschere, Bert, E-mail: bjdebus@sandia.gov and Najm, Habib N., E-mail: hnnajm@sandia.gov},

abstractNote = {Direct solutions of the Chemical Master Equation (CME) governing Stochastic Reaction Networks (SRNs) are generally prohibitively expensive due to excessive numbers of possible discrete states in such systems. To enhance computational efficiency we develop a hybrid approach where the evolution of states with low molecule counts is treated with the discrete CME model while that of states with large molecule counts is modeled by the continuum Fokker–Planck equation. The Fokker–Planck equation is discretized using a 2nd order finite volume approach with appropriate treatment of flux components. The numerical construction at the interface between the discrete and continuum regions implements the transfer of probability reaction by reaction according to the stoichiometry of the system. The performance of this novel hybrid approach is explored for a two-species circadian model with computational efficiency gains of about one order of magnitude.},

doi = {10.1016/J.JCP.2014.10.026},

journal = {Journal of Computational Physics},

number = ,

volume = 281,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}