Low-rank method for fast solution of generalized Smoluchowski equations

Abstract

An efficient method for the fast solution of a new class of generalized Smoluchowski equations is developed. In contrast to conventional Smoluchowski equations with constant rate coefficients that describe the aggregation kinetics, the rate coefficients of the generalized Smoluchowski equations depend on time. The evolution of these coefficients is governed by another set of equations coupled with the aggregation equations. The most prominent example of such processes is the ballistic aggregation when aggregates of different sizes have different partial temperatures. The conventional approaches fail to deal efficiently with these systems when the number of equations N is large. The new method overcomes this difficulty. It exploits the low-rank approach and drastically decreases computation time: instead of O(N2) operations per time step, the new method requires only O(Nr(r+log⁡N)) operations, where r is the system rank. General techniques for an efficient solution of very large systems of ODE are discussed in the context of the new approach.