TY - JOUR

T1 - Connecting the dots

T2 - Semi-analytical and random walk numerical solutions of the diffusion-reaction equation with stochastic initial conditions

AU - Paster, Amir

AU - Bolster, Diogo

AU - Benson, David A.

N1 - Funding Information:
We thank the two anonymous reviewers for their valuable remarks on the manuscript. A.P. and D.B. would like to express thanks for financial support via Army Office of Research grant W911NF1310082 and NSF grant EAR-1113704. D.A.B. was supported by NSF grants DMS–0539176 and EAR–0749035 . Any opinions, findings, conclusions, or recommendations do not necessarily reflect the views of the funding agencies. This research was supported in part by the Notre Dame Center for Research Computing .

PY - 2014/4/15

Y1 - 2014/4/15

N2 - We study a system with bimolecular irreversible kinetic reaction A + B → θ where the underlying transport of reactants is governed by diffusion, and the local reaction term is given by the law of mass action. We consider the case where the initial concentrations are given in terms of an average and a white noise perturbation. Our goal is to solve the diffusion-reaction equation which governs the system, and we tackle it with both analytical and numerical approaches. To obtain an analytical solution, we develop the equations of moments and solve them approximately. To obtain a numerical solution, we develop a grid-less Monte Carlo particle tracking approach, where diffusion is modeled by a random walk of the particles, and reaction is modeled by annihilation of particles. The probability of annihilation is derived analytically from the particles' co-location probability. We rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number. This enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies.

AB - We study a system with bimolecular irreversible kinetic reaction A + B → θ where the underlying transport of reactants is governed by diffusion, and the local reaction term is given by the law of mass action. We consider the case where the initial concentrations are given in terms of an average and a white noise perturbation. Our goal is to solve the diffusion-reaction equation which governs the system, and we tackle it with both analytical and numerical approaches. To obtain an analytical solution, we develop the equations of moments and solve them approximately. To obtain a numerical solution, we develop a grid-less Monte Carlo particle tracking approach, where diffusion is modeled by a random walk of the particles, and reaction is modeled by annihilation of particles. The probability of annihilation is derived analytically from the particles' co-location probability. We rigorously derive the relationship between the initial number of particles in the system and the amplitude of white noise represented by that number. This enables us to compare the particle simulations and the approximate analytical solution and offer an explanation of the late time discrepancies.

KW - Bimolecular reaction

KW - Diffusion-reaction equation

KW - Incomplete mixing

KW - Particle methods

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=84893445379&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2014.01.020

DO - 10.1016/j.jcp.2014.01.020

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AN - SCOPUS:84893445379

SN - 0021-9991

VL - 263

SP - 91

EP - 112

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -