Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique
Provides theoretical convergence guarantees for a widely used numerical method in population balance modeling, relevant to fields like aerosol physics and polymer science.
The paper analyzes the convergence of the fixed pivot technique for solving nonlinear aggregation population balance equations, finding second-order convergence on uniform and non-uniform smooth meshes, first-order on locally uniform meshes, and no convergence on oscillatory or random meshes.
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.