Connecting the dots: Semianalytical and random walk numerical solutions of the diffusion–reaction equation with stochastic initial conditions
We study a system with bimolecular irreversible kinetic reaction A+B→∅ 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 gridless 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' colocation 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.  Graphical abstract:.
 Authors:

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 Environmental Fluid Mechanics Laboratories, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN (United States)
 (Israel)
 Hydrologic Science and Engineering, Colorado School of Mines, Golden, CO, 80401 (United States)
 Publication Date:
 OSTI Identifier:
 22314858
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 263; 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; ANALYTICAL SOLUTION; APPROXIMATIONS; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; CONCENTRATION RATIO; DIFFUSION; DIFFUSION EQUATIONS; DISTURBANCES; GRAPH THEORY; MONTE CARLO METHOD; NOISE; NUMERICAL SOLUTION; PARTICLES; PROBABILITY; RANDOMNESS; STOCHASTIC PROCESSES