Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique

TL;DR

This paper analyzes the convergence properties of the fixed pivot technique for solving nonlinear aggregation population balance equations, demonstrating second order convergence on smooth meshes and limitations on irregular meshes.

Contribution

It provides a detailed convergence analysis of the fixed pivot technique across various mesh types, highlighting its accuracy and limitations.

Findings

Second order convergence on uniform and smooth non-uniform meshes

First order convergence on locally uniform meshes

Lack of convergence on oscillatory and random meshes

Abstract

In this paper, we introduce the convergence analysis of the fixed pivot technique given by S.Kumar and Ramkrishna \cite{Kumar:1996-1} for the nonlinear aggregation population balance equations which are of substantial interest in many areas of science: colloid chemistry, aerosol physics, astrophysics, polymer science, oil recovery dynamics, and mathematical biology. In particular, we investigate the convergence for five different types of uniform and non-uniform meshes which turns out that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Moreover, it yields first order convergence on a locally uniform mesh. Finally, the analysis exhibits that the method does not converge on an oscillatory and non-uniform random meshes. Mathematical results of the convergence analysis are also demonstrated numerically.