Newton method for stationary and quasi-stationary problems for Smoluchowski-type equations

An efficient implementation of the Newton–Krylov iterative method finding numerical solutions of aggregation–fragmentation equations without spontaneous fragmentation is reported. It is based on application of fast numerical schemes for evaluation of the Smoluchowski operator associated with these e...

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Bibliographic Details
Published in:	Journal of computational physics Vol. 382; pp. 124 - 137
Main Authors:	Timokhin, I.V., Matveev, S.A., Siddharth, N., Tyrtyshnikov, E.E., Smirnov, A.P., Brilliantov, N.V.
Format:	Journal Article
Language:	English
Published:	Cambridge Elsevier Inc 01-04-2019 Elsevier Science Ltd
Subjects:	Agglomeration Aggregation–fragmentation kinetics Computational physics Fragmentation Iterative methods Low-rank matrices Mathematical models Matrix methods Multiplication Newton method Newton methods Preconditioners Smoluchowski equations Preconditioners Aggregation–fragmentation kinetics Newton method Smoluchowski equations Low-rank matrices
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Summary:	An efficient implementation of the Newton–Krylov iterative method finding numerical solutions of aggregation–fragmentation equations without spontaneous fragmentation is reported. It is based on application of fast numerical schemes for evaluation of the Smoluchowski operator associated with these equations, and structure of Jacobi matrix, allowing implementation of fast multiplication by a vector. We also propose an efficient preconditioner based on lower triangular band matrix extracted from a Jacobi matrix, resulting in a dramatic speedup of internal GMRES-based iterations between the Newton steps. We compare the efficiency of the novel method with the existing ones and demonstrate a noticeable superiority of our approach. Moreover, we show that the new method may be utilized for modeling of a wider class of aggregation–fragmentation phenomena and derive a new class of quasi-stationary solutions for aggregation–fragmentation equations source of monomers. •Efficient implementation of Newton–Krylov method for (quasi-)stationary problems for Smoluchowski-type equations.•Fast algorithm for directional derivatives of non-linear Smoluchowski-type operator.•An effective preconditioner for Jacobian of Smoluchowski-type operator.
ISSN:	0021-9991 1090-2716
DOI:	10.1016/j.jcp.2019.01.013