Connecting the dots: Semi-analytical and random walk numerical solutions of the diffusion–reaction equation with stochastic initial conditions

Journal Article · Tue Apr 15 00:00:00 EDT 2014 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2014.01.020· OSTI ID:22314858

Paster, Amir ^[1]; Bolster, Diogo ^[1]; Benson, David A. ^[2]

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1. Environmental Fluid Mechanics Laboratories, Dept. of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, Notre Dame, IN (United States)
2. Hydrologic Science and Engineering, Colorado School of Mines, Golden, CO, 80401 (United States)

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 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. - Graphical abstract:.

• OSTI ID: 22314858
• Journal Information: Journal of Computational Physics, Vol. 263; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

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