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

• Published in: Journal of computational physics Vol. 282; pp. 23 - 32
• Main Authors: Matveev, S.A., Smirnov, A.P., Tyrtyshnikov, E.E.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-02-2015
• Subjects: Approximation; Cauchy problem; Cross matrix interpolations; Discretization; Fast algorithms of linear algebra; Finite difference method; Kernels; Mathematical analysis; Predictor-corrector methods; Predictor–corrector scheme; Quadratures; Smoluchowski coagulation equation; Smoluchowski coagulation equation; Fast algorithms of linear algebra; Predictor–corrector scheme; Cross matrix interpolations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.11.003

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.