Stochastic simulation of reactiondiffusion systems: A fluctuatinghydrodynamics approach
Here, we develop numerical methods for stochastic reactiondiffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reactiondiffusion systems we consider, our model becomes similar to the reactiondiffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHDbased description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steadystate fluctuations for the linearized reactiondiffusion model, we construct several twostage (predictorcorrector) schemes, where diffusion is treated using a stochastic CrankNicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly secondorder tau leapingmore »
 Authors:

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 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
 San Jose State Univ., CA (United States). Dept. of Physics and Astronomy
 New York Univ. (NYU), NY (United States). Courant Inst. of Mathematical Sciences
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231; SC0008271
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Chemical Physics
 Additional Journal Information:
 Journal Volume: 146; Journal Issue: 12; Journal ID: ISSN 00219606
 Publisher:
 American Institute of Physics (AIP)
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
 OSTI Identifier:
 1379783
Kim, Changho, Nonaka, Andy, Bell, John B., Garcia, Alejandro L., and Donev, Aleksandar. Stochastic simulation of reactiondiffusion systems: A fluctuatinghydrodynamics approach. United States: N. p.,
Web. doi:10.1063/1.4978775.
Kim, Changho, Nonaka, Andy, Bell, John B., Garcia, Alejandro L., & Donev, Aleksandar. Stochastic simulation of reactiondiffusion systems: A fluctuatinghydrodynamics approach. United States. doi:10.1063/1.4978775.
Kim, Changho, Nonaka, Andy, Bell, John B., Garcia, Alejandro L., and Donev, Aleksandar. 2017.
"Stochastic simulation of reactiondiffusion systems: A fluctuatinghydrodynamics approach". United States.
doi:10.1063/1.4978775. https://www.osti.gov/servlets/purl/1379783.
@article{osti_1379783,
title = {Stochastic simulation of reactiondiffusion systems: A fluctuatinghydrodynamics approach},
author = {Kim, Changho and Nonaka, Andy and Bell, John B. and Garcia, Alejandro L. and Donev, Aleksandar},
abstractNote = {Here, we develop numerical methods for stochastic reactiondiffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reactiondiffusion systems we consider, our model becomes similar to the reactiondiffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHDbased description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steadystate fluctuations for the linearized reactiondiffusion model, we construct several twostage (predictorcorrector) schemes, where diffusion is treated using a stochastic CrankNicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly secondorder tau leaping method. We find that an implicit midpoint tau leaping scheme attains secondorder weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reactiondiffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamic fluctuations to the formation of a twodimensional Turinglike pattern and examine the effect of fluctuations on threedimensional chemical front propagation. Furthermore, by comparing stochastic simulations to deterministic reactiondiffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.},
doi = {10.1063/1.4978775},
journal = {Journal of Chemical Physics},
number = 12,
volume = 146,
place = {United States},
year = {2017},
month = {3}
}