Application of the finite element method to the multicomponent general dynamic equation of aerosols

This paper focuses on the numerical approximation of the multicomponent dynamic equation of aerosols (MCGDE), an integro-partial differential equation that describes the temporal evolution of the aerosol number distribution containing several compounds. More specifically, we compare the performance of the finite element method (FEM) and a commonly used sectional method. In addition to the standard Galerkin FEM, we introduce the Petrov-Galerkin Finite Element Method (PGFEM) to the MCGDE. These methods are compared to analytical solutions of the MCGDE and to accurate solutions given by the discrete multicomponent GDE which models the condensation process numerically at the monomer level. Results show that in cases where condensation is the dominating process, the accuracy of the FEM and PGFEM can be significantly better than the accuracy of the sectional method. They also generally achieve a given accuracy of the solution with a significantly lower number of discretization points than the sectional method. Moreover, the sparser the discretization is, the more beneficial is the use of the PGFEM, because it stabilizes the solution of condensation dominated MCGDEs. In the case of pure coagulation, all three numerical methods yield a similar accuracy.