Random element method for numerical modeling of diffusional processes
The random element method is a generalization of the random vortex method that was developed for the numerical modeling of momentum transport processes as expressed in terms of the Navier-Stokes equations. The method is based on the concept that random walk, as exemplified by Brownian motion, is the stochastic manifestation of diffusional processes. The algorithm based on this method is grid-free and does not require the diffusion equation to be discritized over a mesh, it is thus devoid of numerical diffusion associated with finite difference methods. Moreover, the algrithm is self-adaptive in space and explicit in time, resulting in an improved numerical resolution of gradients as well as a simple and efficient computational procedure.
The method is applied here to an assortment of problems of diffusion of momentum and energy in one-dimension as well as heat conduction in two-dimensions in order to assess its validity and accuracy. The numerical solutions obtained are found to be in good agreement with exact solution except for a statistical error introduced by using a finite number of elements, the error can be reduced by increasing the number of elements or by using ensemble averaging over a number of solutions.
KeywordsRandom Walk Diffusion Equation Couette Flow Dirac Delta Function Heat Conduction Problem
Unable to display preview. Download preview PDF.
- 1.Chorin, A.J. (1973) “Numerical Studies of Slightly Viscous Flow,” J. Fluid Mech., 57, 785–796.Google Scholar
- 3.Chorin, A.J. (1980) “Vortex Models and Boundary Layer Instability,” SIAM J. Scientific Stat Comp., 1, 1–24.Google Scholar
- 4.Einstein, A. (1926) Investigation on the Theory of the Brownian Movement. Translation, Methuen and Co., Ltd., London: (Reprint, Dover Publications, Inc., New York, 1956).Google Scholar
- 5.Courant, R., Friedrichs, K., and Levy, H. (1928) On the Partial Difference Equations of Mathematical Physics (Translation from Mathematische Annalen, 100) AEC Computing Facility, Institute of Mathematical Sciences, New York University, 1956.Google Scholar
- 6.Ghoniem, A.F., Chorin, A.J. and Oppenheim, A.K., “Numerical Modeling of Turbulent Flow in a Combustion Tunnel” Phil. Trans. Royal Soc. Lond., A 304, 303–325.Google Scholar