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In this work we present a parallel implementation of numerical algorithm solving the Cauchy problem for equation of advection of coagulating particles. This equation describes time-evolution of the concentration f (x, v, t) of particles of size v at the point x at the time-moment t. Our numerical algorithm is based on use of total variation diminishing (TVD) scheme and perfectly matching layers (PML) for approximation of advection operator along spatial coordinate x and utilization of the fast numerical method for evaluation of coagulation integrals exploiting low-rank decomposition of coagulation kernel coefficients and fast FFT-based implementation of convolution operation along particle size coordinate v. In our work we exploit one-dimensional domain decomposition approach along spatial coordinate x because it allows to avoid use of parallel FFT implementations which are very expensive in terms of data exchanges and have poor parallel scalability. Moreover, locality of finite-difference operator from TVD-scheme along x coordinate allows to obtain good scalability even for computing clusters with slow network interconnect due to modest volumes of data necessary for synchronization exchanges between times integration steps.

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We explore analytically and numerically agglomeration driven by advection and localized source. The system is inhomogeneous in one dimension, viz. along the direction of advection. It is characterised by the kinetic coefficients – the advection velocity, diffusion coefficient and the reaction kernel, quantifying the aggregation rates. We analyze a simplified model with mass-independent advection velocity, diffusion coefficient, and reaction rates. We also examine a model with mass-dependent coefficients arising in the context of aggregation with sedimentation. For the quasi-stationary case and simplified model, we obtain an exact solution for the spatially dependent agglomerate densities. For the case of mass dependent coefficients we report a new conservation law and develop a scaling theory for the densities. For the numerical efficiency we exploit the low-rank approximation technique; this dramatically increases the computational speed and allows simulations of very large systems. The numerical results are in excellent agreement with the predictions of our theory.

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