A Monte Carlo method for scalar reaction diffusion equations
SIAM Journal on Scientific and Statistical Computing
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
SIAM Journal on Scientific Computing
Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences
Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences
Hi-index | 31.45 |
We study a system with bimolecular irreversible kinetic reaction A+B-@A 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.