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 discretizati...

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Bibliographic Details
Published in:Journal of computational physics 2015-02, Vol.282, p.23-32
Main Authors: Matveev, S.A., Smirnov, A.P., Tyrtyshnikov, E.E.
Format: Article
Language:English
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
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2014.11.003