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.