Aerosol coagulation is a significant process regarding the dynamics of aerosols in the atmosphere. This process leads to a modification of the size distribution of aerosols over time, induced by particle collisions. Aerosol coagulation is described by Smoluchowski equation [1], a partial integral differential equation with introduces non-locality over the aerosol volume variable. Providing a large number of degrees of freedom to generate accurate simulations is computationally demanding for typical discretisation methods [2] for this kind of process.
We investigate the effectiveness of the reduced order basis method [3] to provide accurate and computationally less demanding simulations, by constructing a reduced order subspace included within the reference high-dimensional space used to provide high fidelity simulations.
In the context of typical urban aerosol concentrations [4], we observe that an exponential decay in Kolmogorov n-width enabling significant dimensionality reduction while retaining accuracy. We also provide a reduced order dynamical operator, which online complexity is independent of the high-fidelity dimension. Furthermore, we investigate ways to ensure that some properties of the high-fidelity operator are preserved on reduced subspaces, such as mass conservation and positivity [5]. [1] M. V. Smoluchowski. Drei vortage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Physik, 17, 557–585, 1916 [2] E. Debry, B. Sportisse. Solving aerosol coagulation with size-binning methods. Applied Numerical Mathematics, 57(9), 1008–1020, 2007. [3] A. Quarteroni, et al. Reduced Basis Methods for Partial Differential Equations. Springer Cham. 2015. [4] C. Seigneur, et al. Simulation of aerosol dynamics : A comparative review of mathematical models. Aerosol Science and Technology, 5, 205–222, 1986. [5] F. Filbet, and P. Laurençot. Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation. Archiv der Mathematik, 83(6), 558-567, 2004.