A Reduced Ordel Model for Aerosol Coagulation

Aerosol coagulation is a significant process regarding the dynamics of aerosols in the atmosphere. This process leads to an evolution of the size distribution of aerosols over time induced by particle collisions, and is described by Smoluchowski equation [1].
Accurate numerical simulations of this process are computationally demanding by typical discretization methods [2]. We investigate the effectiveness of the reduced basis method [3] to provide accurate but less demanding simulations, by constructing a reduced order subspace included within the reference high-dimensional space used to provide high fidelity simulations.
We also provide residual-based a posteriori error estimates [4] which enable certification of results up to a given error tolerance. We obtain efficient online model and error estimates as their online computational cost only scale with the reduced dimension, and not the dimension of the high-fidelity model.
By careful design of reduced subspaces we ensure that some properties of the high fidelity operator, such as mass conservation [5], are also preserved by reduced order models.

[1] M. V. Smoluchowski. Drei vortage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen. Physik, 17, 557–585, 1916
[2] E. Debry, and 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] M. Grepl, and A. Patera, A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential EquationsESAIM: Mathematical Modelling and Numerical Analysis, 39(1), 157-181, 2005
[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