Application of finite element method to General Dynamic Equation of Aerosols – Comparison with classical numerical approximations

Abstract

This paper focuses on the numerical approximation of the general dynamic equation of aerosols (GDE), an integro-partial differential equation, which models the temporal evolution of aerosol number distributions. More specifically, we study the applicability of the finite element method (FEM) to numerically approximate the GDE at different conditions. In addition to applying the conventional Galerkin FEM to the GDE, we also introduce its extension, the Petrov–Galerkin FEM to improve the stability of the approximation. The FEM approximations are compared against the so-called sectional method, based on the finite difference approximation, which is commonly used for the numerical approximation of the GDE. The FEM schemes and the sectional method are validated with series of numerical simulations, where an accurate solution of GDE is available. In these simulations, we consider special cases of pure condensation and coagulation, as well as cases including both of these processes. The accurate solutions – to which numerical approximations are compared – consist of either the analytical solutions or an accurate solution to the discrete GDE, where the particle size range is discretized into intervals of the size of a monomer, and the condensation process is modeled as an interaction between aerosol particles and monomers. The results of this paper demonstrate that while in the solution of the coagulation problem, the conventional sectional method is more efficient than the FEM, in cases of the condensation equation and the condensation-dominated GDE, the FE approximations outperform the sectional method – yielding a desired level of accuracy of the solution using less discretization points and in computation time which can be even orders of magnitude lower than with the sectional method.