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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 [3] we solved this problem in the case when there is no di usion. The non-zero di usion term drastically complicates the situation. Under some general assumptions we give the interrelations between the homogeneous and inhomogeneous cases. This provides an e ective 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 e ective source. We give a numerical comparison of these two models. The 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 ames 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 di usion coe cient 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 di usion coe cient depends on the particle's size is open.

Stochastic Lagrangian model for spatially inhomogeneous Smoluchowski equation governing coagulating and diffusing particles