Technical Report 3, c4e-Preprint Series, Cambridge

Authors: Markus Kraft* and Mike J. Goodson

Reference: Technical Report 3, c4e-Preprint Series, Cambridge, 2001

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Nano-particles produced at high temperatures often undergo rapid coalescence with complex associated rate laws. In this paper we develop and study the numerical properties of a stochastic algorithm for the modelling of nano-particle dynamics in the free molecular regime. Following the work in (A. Eibeck and W. Wagner, SIAM J. Sci Comput. 22(3):8022-821,2000), we model the system as a Markov process and introduce a new majorant kernel that enables us to extend the use of fictitious jumps to a wider class of problems. We also include a source term. We study the convergence properties of the algorithm; the systematic error decreases as 1/N. We then examine the efficiency of the algorithm by comparing it to a direct simulation Monte Carlo (DSMC) algorithm, that described by Gillespie (D.T. Gillespie, J. Atmos. Sci. 32(10):1977-1989, 1975). We also compare the efficiency of our new majorant with the linear majorant used in (A. Eibeck and W. Wagner, Ann. Appl. Prob. 11(4):1137-1165,2001). The improved stochastic algorithm compares very favourably with the DSMC algorithm. The CPU time required for simulation is orders of magnitude lower, and for low particle numbers, the CPU time increases linearly with particle number, rather than as the square of the particle number (as with the DSMC algorithm). Our majorant kernel enables us to simulate solutions to a wider variety of problems than the linear majorant and also gives a significant gain in efficiency. The results of this report promise excellent efficiency of simulation of problems such as soot formation and synthesis of fumed silica, and also for extension to a more general class of problems in which the population balance equation occurs.

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