An efficient particle tracking equation with specified spatial step for the solution of the diffusion equation

Abstract

The traditional di(usive particle tracking equation provides an updated particle location as a function of its previous location and molecular di(usion coe cient while maintaining a constant time step. A smaller time step yields an increasingly accurate, yet more computationally demanding solution. Selection of this time step becomes an important consideration and, depending on the complexity of the problem, a single optimum value may not exist. The characteristics of the system under consideration may be such that a constant time step may yield solutions with insu cient accuracy in some portions of the domain and excess computation time for others. In this work, new particle tracking equations speci%cally designed for the solution of problems associated with di(usion processes in one, two, and three dimensions are presented. Instead of a constant time step, the proposed equations employ a constant spatial step. Using a traditional particle tracking algorithm, the travel time necessary for a particle with a di(usion coe cient inversely proportional to its diameter to achieve a pre-speci%ed distance was determined. Because the size of a particle a(ects how it di(uses in a quiescent 8uid, it is expected that two di(erently sized particles would require di(erent travel times to move a given distance. The probability densities of travel times for plumes of monodisperse particles were consistently found to be log-normal in shape. The parameters describing this log-normal distribution, i.e., mean and standard deviation, are functions of the distance speci%ed for travel and the di(usion coe cient of the particles. Thus, a random number selected from this distribution approximates the time required for a given particle to travel a speci%ed distance. The new equations are straightforward and may be easily incorporated into appropriate particle tracking algorithms. In addition, the new particle tracking equations are as accurate and often more computationally e cient than the traditional particle tracking equation