Optimal approximation of harmonic growth clusters by orthogonal polynomials

TL;DR

This paper discusses an approximation scheme for interface dynamics in two-dimensional systems with conservation laws, using orthogonal polynomials with a deformed Gaussian kernel, relevant to various physical processes.

Contribution

It introduces an approximation method based on orthogonal polynomials with a deformed Gaussian kernel for modeling boundary evolution in complex physical systems.

Findings

Provides an efficient approximation scheme for interface dynamics

Establishes relations to potential theory

Applicable to various physical processes like fluid flows and aggregation

Abstract

Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of Hele-Shaw flows, [1]), to the granular dynamics of hard spheres [2], and even diffusion-limited aggregation [3]. Although a complete solution for the continuum case exists [4, 5], efficient approximations of the boundary evolution are very useful due to their practical applications [6]. In this article, the approximation scheme based on orthogonal polynomials with a deformed Gaussian kernel [7] is discussed, as well as relations to potential theory.