Coagulation, non-associative algebras and binary trees

TL;DR

This paper introduces a novel algebraic and combinatorial approach to solving the Smoluchowski coagulation equation using non-associative algebras and binary trees, leading to a new efficient numerical simulation method.

Contribution

It develops a deterministic solution expansion in a non-associative algebra framework and links it to binary trees, providing a new numerical method with linear-logarithmic complexity.

Findings

Solution expansion represented by binary trees

Compatibility of binary-tree generating procedures established

New fast Fourier transform-based numerical simulation method

Abstract

We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by: (i) grafting together the roots of all pairs of order-compatibile trees at preceding orders, or (ii) attaching binary branches to all free branches of trees at the previous order. We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.