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Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation

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  • We introduce a general family of Weighted Flow Algorithms for simulating particle coagulation, generate a method to optimally tune these methods, and prove their consistency and convergence under general assumptions. These methods are especially effective when the size distribution of the particle population spans many orders of magnitude, or in cases where the concentration of those particles that significantly drive the population evolution is small relative to the background density. We also present a family of simulations demonstrating the efficacy of the method.

    Mathematics Subject Classification: Primary: 65C35, 60K35; Secondary: 60J28.


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  • Figure 1.  Size distributions for two particle sub-populations using the scheme described in Section 6.1 with $ N_{\rm p} = 10^3 $ particles. Top: equal weightings for both distributions. Bottom: equal computational number for each distribution. The points are the mean of the particle process, while the error bars show 95% spread of realizations, not confidence intervals for the mean. The solid lines are very accurate finite-volume solutions. Using an equal-number rather than equal-weight weighting reduces sub-population 2 expected error by about 17 times, with an increase in sub-population 1 expected error of only about 1.4 times

    Figure 2.  Errors for two particle sub-populations for varying number of particles $ N_{\rm p} $. The weight ratio $ r = w^1/w^2 $ is $ 1 $, $ 10 $, $ 10^2 $, $ 10^3 $, $ 10^4 $, $ 10^5 $, and $ 10^6 $, moving from upper-left to down and to the right on each line. The circle points correspond to equal weights for both sub-populations (top panel in Figure 1), while the square points correspond to equal numbers of computational particles for each sub-population (bottom panel in Figure 1). Error bars are not shown as they are visually negligible

    Figure 3.  Convergence of the size distribution using the weighting scheme from Section 6.1 for different weight ratios $ r $ for the distribution of group $ a = 1 $ particles and group $ a = 2 $ particles. The baseline for computing the error is a very accurate finite-volume solution $ n_{\rm fv} $

    Figure 4.  Size distributions (left column: number, right column: mass) for a sedimentation kernel computed using three different weighting schemes and $ N_{\rm p} = 10^3 $ particles. Top row: flat weighting. Center row: inverse-mass weighting. Bottom row: Combined flat-and-inverse-mass weighting as described in Section 6.2. The blue and red circles show the particle solution at times $ t = 0 $ and $ t = 10\rm\ min $, respectively, while the solid lines are very accurate finite-volume solutions. Observe that the combined weighting scheme accurately captures both the number and mass size distributions

    Figure 5.  Number and mass errors for the sedimentation simulations described in Section 6.2 shown in Figure 4 for varying number of particles $ N_{\rm p} $. The solid lines show size-weighted simulations with varying weight exponent $ \alpha \in \{0, -1, -2, -3\} $ (ordered from top to bottom). The filled circles are combined flat-and-inverse-mass weighted simulations

    Figure 6.  Convergence of the number and mass size distribution using the weighting scheme from Section 6.2 for different weight exponents $ \alpha $ and for the combined weighting. The baseline for computing the error is a very accurate finite-volume solution with number distribution $ n_{\rm fv} $ and mass distribution $ m_{\rm fv} $. The exponents $ \alpha = -1 $ and $ \alpha = -2 $ have similar behavior (not plotted)

    Figure 7.  Size distributions (left column: number, right column: mass) for a Brownian kernel simulation with two particle classes (blue and red), with the same parameters as in Figure 1. Top row: flat size weighting ($ \alpha = 0 $) with equal weights in each class ($ r = 1 $). Second row: flat size weighting ($ \alpha = 0 $) with equal number in each class ($ r = 10^3 $). Third row: combined $ \alpha = 0 $ and $ \alpha = -3 $ size weighting with equal weights in each class ($ r = 1 $). Bottom row: combined $ \alpha = 0 $ and $ \alpha = -3 $ size weighting with equal number in each class ($ r = 10^3 $)

    Table 1.  Notation and variable ranges

    Variables Meaning Range
    $ a,b,c,d $ physical particle class $ \mathbb{M} = \{1,2,3,\ldots,M\} $
    $ \alpha,\beta,\gamma $ event outcomes $ \{0,1\} $
    $ e $ basis vector $ \mathbb{M} \times \mathbb{N}_1 \to \mathbb{Z} $
    $ i,j,k $ physical particle size $ \mathbb{N}_1 = \{1,2,3,\ldots\} $
    $ K $ coagulation kernel $ [0,\infty) $
    $ \lambda $ event rate $ [0,\infty) $
    $ N $ physical number concentration $ [0,\infty) $
    $ N_{\rm p} $ total number of computational particles $ \mathbb{N}_0 = \{0,1,2,\ldots\} $
    $ p $ probability $ [0,1] $
    $ q,Q $ number of computational particles $ \mathbb{N}_0 = \{0,1,2,\ldots\} $
    $ \rho $ event rate $ [0,\infty) $
    $ T $ selection rate function $ [0,\infty) $
    $ V $ computational volume $ (0,\infty) $
    $ w $ weighting function $ (0,\infty) $
    $ x,X $ physical particle concentration $ [0,\infty) $
    $ y,Y $ computational particle concentration $ [0,\infty) $
    $ \zeta $ event jump $ \mathbb{M} \times \mathbb{N}_1 \to \mathbb{Z} $
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