A fast numerical method for the Cauchy problem for the Smoluchowski equation

A new solution technique is proposed for one-dimensional Smoluchowski equations. It is based on the finite-difference predictor-corrector scheme and is faster than other methods using this kind of scheme. The new technique capitalizes on low-rank approximations of matrices arising after discretization of the coagulation kernel and includes a new fast convolution algorithm with the trapezoidal quadrature rule. For the grids with N nodes, the complexity of the new method is O(Nlog⁡N) for each step with time instead of O(N2) operations required by the standard implementation of the predictor-corrector scheme.