An optimal scaling to computationally tractable dimensionless models: Study of latex particles morphology formation

TL;DR

This paper introduces a new optimal scaling method to transform complex models with large coefficient variations into more manageable dimensionless forms, improving numerical stability and physical plausibility.

Contribution

The paper presents a novel Optimal Scaling approach that reduces coefficient variation in models, demonstrated on Latex Particle Morphology formation, enhancing computational tractability.

Findings

Coefficient variation reduced from 49 to 4 orders of magnitude.

Improved numerical accuracy and stability in solving the PBE.

Method applicable across various scientific modeling fields.

Abstract

In modelling of chemical, physical or biological systems it may occur that the coefficients, multiplying various terms in the equation of interest, differ greatly in magnitude, if a particular system of units is used. Such is, for instance, the case of the Population Balance Equations (PBE) proposed to model the Latex Particles Morphology formation. The obvious way out of this difficulty is the use of dimensionless scaled quantities, although often the scaling procedure is not unique. In this paper, we introduce a conceptually new general approach, called Optimal Scaling (OS). The method is tested on the known examples from classical and quantum mechanics, and applied to the Latex Particles Morphology model, where it allows us to reduce the variation of the relevant coefficients from 49 to just 4 orders of magnitudes. The PBE are then solved by a novel Generalised Method Of Characteristics, and the OS is shown to help reduce numerical error, and avoid unphysical behaviour of the solution. Although inspired by a particular application, the proposed scaling algorithm is expected find application in a wide range of chemical, physical and biological problems.