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

Abstract

The traditional diffusive particle tracking equation provides an updated particle location as a function of its previous location and molecular diffusion coefficient 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 insufficient accuracy in some portions of the domain and excess computation time for others. In this work, new particle tracking equations specifically designed for the solution of problems associated with diffusion 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 diffusion coefficient inversely proportional to its diameter to achieve a pre-specified distance was determined. Because the size of a particle affects how it diffuses in a quiescent fluid, it is expected that two differently sized particles would require different 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 specified for travel and the diffusion coefficient of the particles. Thus, a random number selected from this distribution approximates the time required for a given particle to travel a specified 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 efficient than the traditional particle tracking equation.