Stochastic Lagrangian models and algorithms for spatially inhomogeneous Smoluchowski equation

The following generally unsolved yet problem is studied: construct the solution of a spatially inhomogeneous Smoluchowski equation governing coagulating and diffusing particles in a host gas, on the basis of solutions to homogeneous Smoluchowski equation. In [Math. Comput. Simul. 49 (1999) 57], we solved this problem in the case when there is no diffusion. The non-zero diffusion term drastically complicates the situation. Under some general assumptions we give the interrelations between the homogeneous and inhomogeneous cases. This provides an effective numerical scheme especially when the host gas is incompressible. New Lagrangian scheme leads to a new model governing by a Smoluchowski type equation with an additional effective source. We give a numerical comparison of these two models. dThe computer time of the new algorithm is so dramatically decreased, compared to the conventional deterministic algorithm (tens of hours drop down to several minutes) that many practical problems like the formation of soot particles in flames or chemical reactions coupled to formation of a new phase can be solved in a reasonable computer time. However, this method works only if the diffusion coefficient of all particles is the same which can be a reasonable approximation only for special systems. The problem of generalisation of the method presented to the case when the diffusion coefficient depends on the particle's size is open. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.