Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation

Lee DeVille; Nicole Riemer; Matthew West

doi:10.3934/jcd.2019003

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June 2019, 6(1): 69-94. doi: 10.3934/jcd.2019003

Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation

Lee DeVille ^1, , Nicole Riemer ^2, and Matthew West ^3,

1.	Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA
2.	Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, 1301 W. Green Street, Urbana, IL 61801, USA
3.	Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 W. Green Street, Urbana, IL 61801, USA

Published March 2019

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.

Keywords: Stochastic simulation, Martingales, particle coagulation, Smoluchowski equation.

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

Citation: Lee DeVille, Nicole Riemer, Matthew West. Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation. Journal of Computational Dynamics, 2019, 6 (1) : 69-94. doi: 10.3934/jcd.2019003

References:

[1]	H. Babovsky, On a Monte Carlo scheme for Smoluchowski's coagulation equation, Monte Carlo Methods and Appl., 5 (1999), 1-18. doi: 10.1515/mcma.1999.5.1.1.
[2]	K. V Beard, Terminal velocity and shape of cloud and precipitation drops aloft, J. Atmos. Sci., 33 (1976), 851-864. doi: 10.1175/1520-0469(1976)033<0851:TVASOC>2.0.CO;2.
[3]	A. Bott, A flux method for the numerical solution of the stochastic collection equation, J. Atmos. Sci., 55 (1998), 2284-2293. doi: 10.1175/1520-0469(1998)055<2284:AFMFTN>2.0.CO;2.
[4]	J. H. Curtis, M. D. Michelotti, N. Riemer, M. Heath and M. West, Accelerated simulation of stochastic particle removal processes in particle-resolved aerosol models, J. Comput. Phys., 322 (2016), 21-32. doi: 10.1016/j.jcp.2016.06.029.
[5]	M. H. A. Davis, Markov Models and Optimization, Chapman and Hall, Boundary Row, London, 1993. doi: 10.1007/978-1-4899-4483-2.
[6]	E. Debry, B. Sportisse and B. Jourdain, A stochastic approach for the numerical simulation of the general dynamics equations for aerosols, J. Comput. Phys., 184 (2003), 649-669. doi: 10.1016/S0021-9991(02)00041-4.
[7]	L. DeVille, N. Riemer and M. West, Weighted flow algorithms (WFA) for stochastic particle coagulation, J. Comput. Phys., 230 (2011), 8427-8451. doi: 10.1016/j.jcp.2011.07.027.
[8]	J. L. Doob, Stochastic Processes, Wiley Classics Library. John Wiley & Sons Inc., New York, 1990. ISBN 0-471-52369-0. Reprint of the 1953 original, A Wiley-Interscience Publication.
[9]	Y. Efendiev and M. R. Zachariah, Hybrid Monte Carlo method for simulation of two-component aerosol coagulation and phase segregation, J. Colloid Interf. Sci., 249 (2002), 30-43. doi: 10.1006/jcis.2001.8114.
[10]	Y. Efendiev, H. Struchtrup, M. Luskin and M. R. Zachariah, A hybrid sectional-moment model for coagulation and phase segregation in binary liquid nanodroplets, J. Nanopart. Res., 4 (2002), 61-72. doi: 10.1023/A:1020122403428.
[11]	A. Eibeck and W. Wagner, An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena, SIAM J. Sci. Comput., 22 (2000), 802-821. doi: 10.1137/S1064827599353488.
[12]	A. Eibeck and W. Wagner, Approximative solution of the coagulation-fragmentation equation by stochastic particle systems, Stochastic Anal. Appl., 18 (2000), 921-948. doi: 10.1080/07362990008809704.
[13]	A. Eibeck and W. Wagner, Stochastic particle approximations for Smoluchoski's coagulation equation, Ann. Appl. Probab., 11 (2001), 1137-1165. doi: 10.1214/aoap/1015345398.
[14]	A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.
[15]	D. T. Gillespie, The stochastic coalescence model for cloud droplet growth, J. Atmos. Sci., 29 (1972), 1496-1510. doi: 10.1175/1520-0469(1972)029<1496:TSCMFC>2.0.CO;2.
[16]	D. T. Gillespie, An exact method for numerically simulating the stochastic coalescence process in a cloud, J. Atmos. Sci., 32 (1975), 1977-1989. doi: 10.1175/1520-0469(1975)032<1977:AEMFNS>2.0.CO;2.
[17]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[18]	D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.
[19]	D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, 1992.
[20]	W. D. Hall, A detailed microphysical model within a two-dimensional dynamic framework: Model description and preliminary results, J. Atmos. Sci., 37 (1980), 2486-2507. doi: 10.1175/1520-0469(1980)037<2486:ADMMWA>2.0.CO;2.
[21]	L. E. Hatch, J. M. Creamean, A. P. Ault, J. D. Surratt, M. N. Chan, J. H. Seinfeld, E. S. Edgerton, Y. Su and K. A. Prather, Measurements of isoprene-derived organosulfates in ambient aerosols by aerosol time-of-flight mass spectrometry-part 1: Single particle atmospheric observations in Atlanta, Environ. Sci. Technol., 45 (2011), 5105-5111. doi: 10.1021/es103944a.
[22]	L. M. Hildemann, G. R. Markowski, M. C. Jones and G. R. Cass, Submicrometer aerosol mass distributions of emissions from boilers, fireplaces, automobiles, diesel trucks, and meat-cooking operations, Aerosol Sci. Technol., 14 (1991), 138-152. doi: 10.1080/02786829108959478.
[23]	M. Hughes, J. K. Kodros, J. R. Pierce, M. West and N. Riemer, Machine learning to predict the global distribution of aerosol mixing state metrics, Atmosphere, 9 (2018), 15. doi: 10.3390/atmos9010015.
[24]	R. Irizarry, Fast Monte Carlo methodology for multivariate particulate systems-Ⅰ: Point ensemble Monte Carlo, Chem. Eng. Sci., 63 (2008), 95-110. doi: 10.1016/j.ces.2007.09.007.
[25]	R. Irizarry, Fast Monte Carlo methodology for multivariate particulate systems-Ⅱ: $\tau$-PEMC, Chem. Eng. Sci., 63 (2008), 111-121. doi: 10.1016/j.ces.2007.09.006.
[26]	M. Z. Jacobson, Fundamentals of Atmospheric Modeling, Cambridge University Press, 2005. doi: 10.1017/CBO9781139165389.
[27]	M. Z. Jacobson, R. P. Turco, E. J. Jensen and O. B. Toon, Modeling coagulation among particles of different composition and size, Atmos. Environ., 28 (1994), 1327-1338. doi: 10.1016/1352-2310(94)90280-1.
[28]	A. Kolodko and K. Sabelfeld, Stochastic particle methods for Smoluchowski coagulation equation: Variance reduction and error estimations, Monte Carlo Methods Appl., 9 (2003), 315-339. doi: 10.1515/156939603322601950.
[29]	A. B. Kostinski and R. A. Shaw, Fluctuations and luck in droplet growth by coalescence, Bull. Amer. Meteor. Soc., 86 (2005), 235-244. doi: 10.1175/BAMS-86-2-235.
[30]	G. Kotalczyk and F. E. Kruis, A Monte Carlo method for the simulation of coagulation and nucleation based on weighted particles and the concepts of stochastic resolution and merging, J. Comput. Phys., 340 (2017), 276-296. doi: 10.1016/j.jcp.2017.03.041.
[31]	T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes Appl., 6 (1977/78), 223-240. doi: 10.1016/0304-4149(78)90020-0.
[32]	A. Maisels, F. E. Kruis and H. Fissan, Direct simulation Monte Carlo for simultaneous nucleation, coagulation, and surface growth in dispersed systems, Chem. Eng. Sci., 59 (2004), 2231-2239. doi: 10.1016/j.ces.2004.02.015.
[33]	R. McGraw and D. L. Wright, Chemically resolved aersol dynamics for internal mixtures by the quadrature method of moments, J. Aerosol Sci., 34 (2003), 189-209. doi: 10.1016/S0021-8502(02)00157-X.
[34]	B. Øksendal, Stochastic Differential Equations, Universitext. Springer-Verlag, Berlin, sixth edition, 2003. ISBN 3-540-04758-1. An introduction with applications. doi: 10.1007/978-3-642-14394-6.
[35]	R. I. A. Patterson, W. Wagner and M. Kraft, Stochastic weighted particle methods for population balance equations, J. Comput. Phys., 230 (2011), 7456-7472. doi: 10.1016/j.jcp.2011.06.011.
[36]	A. Petzold, J. A. Ogren, M. Fiebig, P. Laj, S.-M. Li, U. Baltensperger, T. Holzer-Popp, S. Kinne, G. Pappalardo, N. Sugimoto and et al., Recommendations for reporting "black carbon" measurements, Atmos. Chem. Phys., 13 (2013), 8365-8379. doi: 10.5194/acp-13-8365-2013.
[37]	X. Qin, K. A. Pratt, L. G. Shields, S. M. Toner and K. A. Prather, Seasonal comparisons of single-particle chemical mixing state in Riverside, CA, Atmos. Environ., 59 (2012), 587-596. doi: 10.1016/j.atmosenv.2012.05.032.
[38]	N. Riemer, H. Vogel, B. Vogel and F. Fiedler, Modeling aerosols on the mesoscale $\gamma$, part Ⅰ: Treatment of soot aerosol and its radiative effects, J. Geophys. Res., 108 (2003), 4601. doi: 10.1029/2003JD003448.
[39]	N. Riemer, M. West, R. A. Zaveri and R. C. Easter, Simulating the evolution of soot mixing state with a particle-resolved aerosol model, J. Geophys. Res., (2009), D09202. doi: 10.1029/2008JD011073.
[40]	J. H. Seinfeld and S. Pandis, Atmospheric Chemistry and Physics, Wiley, 2016.
[41]	S. Shima, K. Kusano, A. Kawano, T. Sugiyama and S. Kawahara, The super-droplet method for the numerical simulation of clouds and precipitation: A particle-based and probabilistic microphysics model coupled with a non-hydrostatic model, Q. J. R. Meteorol. Soc., 135 (2009), 1307-1320. doi: 10.1002/qj.441.
[42]	A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Chapman & Hall, London, 1995.
[43]	J. Tian, N. Riemer, M. West, L. Pfaffenberger, H. Schlager and A. Petzold, Modeling the evolution of aerosol particles in a ship plume using PartMC-MOSAIC, Atmos. Chem. Phys., 14 (2014), 5327-5347. doi: 10.5194/acp-14-5327-2014.
[44]	C. G. Wells and M. Kraft, Direct simulation and mass flow stochastic algorithms to solve a sintering-coagulation equation, Monte Carlo Methods Appl., 11 (2005), 175-197. doi: 10.1515/156939605777585980.
[45]	M. West, N. Riemer, J. Curtis, M. Michelotti and J. Tian, PartMC: Particle-resolved Monte-Carlo atmospheric aerosol simulation, version 2.5.0, 2018. doi: 10.5281/zenodo.1490925.
[46]	C. Yoon and R. McGraw, Representation of generally mixed multivariate aerosols by the quadrature method of moments: Ⅱ. Aerosol dynamics, J. Aerosol Sci., 35 (2004), 577-598. doi: 10.1016/j.jaerosci.2003.11.012.
[47]	H. Zhao and C. Zheng, Correcting the Multi-Monte Carlo Method for particle coagulation, Powder Technol., 193 (2009), 120-123. doi: 10.1016/j.powtec.2009.01.019.
[48]	H. Zhao, C. Zheng and M. Xu, Multi-Monte Carlo Method for coagulation and condensation/evaporation in dispersed systems, J. Colloid Interface Sci., 286 (2005), 195-208. doi: 10.1016/j.jcis.2004.12.037.
[49]	H. Zhao, F. E. Kruis and C. Zheng, Reducing statistical noise and extending the size spectrum by applying weighted simulation particles in Monte Carlo simulation of coagulation, Aerosol Sci. Technol., 43 (2009), 781-793. doi: 10.1080/02786820902939708.

show all references

References:

[1]	H. Babovsky, On a Monte Carlo scheme for Smoluchowski's coagulation equation, Monte Carlo Methods and Appl., 5 (1999), 1-18. doi: 10.1515/mcma.1999.5.1.1.
[2]	K. V Beard, Terminal velocity and shape of cloud and precipitation drops aloft, J. Atmos. Sci., 33 (1976), 851-864. doi: 10.1175/1520-0469(1976)033<0851:TVASOC>2.0.CO;2.
[3]	A. Bott, A flux method for the numerical solution of the stochastic collection equation, J. Atmos. Sci., 55 (1998), 2284-2293. doi: 10.1175/1520-0469(1998)055<2284:AFMFTN>2.0.CO;2.
[4]	J. H. Curtis, M. D. Michelotti, N. Riemer, M. Heath and M. West, Accelerated simulation of stochastic particle removal processes in particle-resolved aerosol models, J. Comput. Phys., 322 (2016), 21-32. doi: 10.1016/j.jcp.2016.06.029.
[5]	M. H. A. Davis, Markov Models and Optimization, Chapman and Hall, Boundary Row, London, 1993. doi: 10.1007/978-1-4899-4483-2.
[6]	E. Debry, B. Sportisse and B. Jourdain, A stochastic approach for the numerical simulation of the general dynamics equations for aerosols, J. Comput. Phys., 184 (2003), 649-669. doi: 10.1016/S0021-9991(02)00041-4.
[7]	L. DeVille, N. Riemer and M. West, Weighted flow algorithms (WFA) for stochastic particle coagulation, J. Comput. Phys., 230 (2011), 8427-8451. doi: 10.1016/j.jcp.2011.07.027.
[8]	J. L. Doob, Stochastic Processes, Wiley Classics Library. John Wiley & Sons Inc., New York, 1990. ISBN 0-471-52369-0. Reprint of the 1953 original, A Wiley-Interscience Publication.
[9]	Y. Efendiev and M. R. Zachariah, Hybrid Monte Carlo method for simulation of two-component aerosol coagulation and phase segregation, J. Colloid Interf. Sci., 249 (2002), 30-43. doi: 10.1006/jcis.2001.8114.
[10]	Y. Efendiev, H. Struchtrup, M. Luskin and M. R. Zachariah, A hybrid sectional-moment model for coagulation and phase segregation in binary liquid nanodroplets, J. Nanopart. Res., 4 (2002), 61-72. doi: 10.1023/A:1020122403428.
[11]	A. Eibeck and W. Wagner, An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena, SIAM J. Sci. Comput., 22 (2000), 802-821. doi: 10.1137/S1064827599353488.
[12]	A. Eibeck and W. Wagner, Approximative solution of the coagulation-fragmentation equation by stochastic particle systems, Stochastic Anal. Appl., 18 (2000), 921-948. doi: 10.1080/07362990008809704.
[13]	A. Eibeck and W. Wagner, Stochastic particle approximations for Smoluchoski's coagulation equation, Ann. Appl. Probab., 11 (2001), 1137-1165. doi: 10.1214/aoap/1015345398.
[14]	A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.
[15]	D. T. Gillespie, The stochastic coalescence model for cloud droplet growth, J. Atmos. Sci., 29 (1972), 1496-1510. doi: 10.1175/1520-0469(1972)029<1496:TSCMFC>2.0.CO;2.
[16]	D. T. Gillespie, An exact method for numerically simulating the stochastic coalescence process in a cloud, J. Atmos. Sci., 32 (1975), 1977-1989. doi: 10.1175/1520-0469(1975)032<1977:AEMFNS>2.0.CO;2.
[17]	D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.
[18]	D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.
[19]	D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, 1992.
[20]	W. D. Hall, A detailed microphysical model within a two-dimensional dynamic framework: Model description and preliminary results, J. Atmos. Sci., 37 (1980), 2486-2507. doi: 10.1175/1520-0469(1980)037<2486:ADMMWA>2.0.CO;2.
[21]	L. E. Hatch, J. M. Creamean, A. P. Ault, J. D. Surratt, M. N. Chan, J. H. Seinfeld, E. S. Edgerton, Y. Su and K. A. Prather, Measurements of isoprene-derived organosulfates in ambient aerosols by aerosol time-of-flight mass spectrometry-part 1: Single particle atmospheric observations in Atlanta, Environ. Sci. Technol., 45 (2011), 5105-5111. doi: 10.1021/es103944a.
[22]	L. M. Hildemann, G. R. Markowski, M. C. Jones and G. R. Cass, Submicrometer aerosol mass distributions of emissions from boilers, fireplaces, automobiles, diesel trucks, and meat-cooking operations, Aerosol Sci. Technol., 14 (1991), 138-152. doi: 10.1080/02786829108959478.
[23]	M. Hughes, J. K. Kodros, J. R. Pierce, M. West and N. Riemer, Machine learning to predict the global distribution of aerosol mixing state metrics, Atmosphere, 9 (2018), 15. doi: 10.3390/atmos9010015.
[24]	R. Irizarry, Fast Monte Carlo methodology for multivariate particulate systems-Ⅰ: Point ensemble Monte Carlo, Chem. Eng. Sci., 63 (2008), 95-110. doi: 10.1016/j.ces.2007.09.007.
[25]	R. Irizarry, Fast Monte Carlo methodology for multivariate particulate systems-Ⅱ: $\tau$-PEMC, Chem. Eng. Sci., 63 (2008), 111-121. doi: 10.1016/j.ces.2007.09.006.
[26]	M. Z. Jacobson, Fundamentals of Atmospheric Modeling, Cambridge University Press, 2005. doi: 10.1017/CBO9781139165389.
[27]	M. Z. Jacobson, R. P. Turco, E. J. Jensen and O. B. Toon, Modeling coagulation among particles of different composition and size, Atmos. Environ., 28 (1994), 1327-1338. doi: 10.1016/1352-2310(94)90280-1.
[28]	A. Kolodko and K. Sabelfeld, Stochastic particle methods for Smoluchowski coagulation equation: Variance reduction and error estimations, Monte Carlo Methods Appl., 9 (2003), 315-339. doi: 10.1515/156939603322601950.
[29]	A. B. Kostinski and R. A. Shaw, Fluctuations and luck in droplet growth by coalescence, Bull. Amer. Meteor. Soc., 86 (2005), 235-244. doi: 10.1175/BAMS-86-2-235.
[30]	G. Kotalczyk and F. E. Kruis, A Monte Carlo method for the simulation of coagulation and nucleation based on weighted particles and the concepts of stochastic resolution and merging, J. Comput. Phys., 340 (2017), 276-296. doi: 10.1016/j.jcp.2017.03.041.
[31]	T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes Appl., 6 (1977/78), 223-240. doi: 10.1016/0304-4149(78)90020-0.
[32]	A. Maisels, F. E. Kruis and H. Fissan, Direct simulation Monte Carlo for simultaneous nucleation, coagulation, and surface growth in dispersed systems, Chem. Eng. Sci., 59 (2004), 2231-2239. doi: 10.1016/j.ces.2004.02.015.
[33]	R. McGraw and D. L. Wright, Chemically resolved aersol dynamics for internal mixtures by the quadrature method of moments, J. Aerosol Sci., 34 (2003), 189-209. doi: 10.1016/S0021-8502(02)00157-X.
[34]	B. Øksendal, Stochastic Differential Equations, Universitext. Springer-Verlag, Berlin, sixth edition, 2003. ISBN 3-540-04758-1. An introduction with applications. doi: 10.1007/978-3-642-14394-6.
[35]	R. I. A. Patterson, W. Wagner and M. Kraft, Stochastic weighted particle methods for population balance equations, J. Comput. Phys., 230 (2011), 7456-7472. doi: 10.1016/j.jcp.2011.06.011.
[36]	A. Petzold, J. A. Ogren, M. Fiebig, P. Laj, S.-M. Li, U. Baltensperger, T. Holzer-Popp, S. Kinne, G. Pappalardo, N. Sugimoto and et al., Recommendations for reporting "black carbon" measurements, Atmos. Chem. Phys., 13 (2013), 8365-8379. doi: 10.5194/acp-13-8365-2013.
[37]	X. Qin, K. A. Pratt, L. G. Shields, S. M. Toner and K. A. Prather, Seasonal comparisons of single-particle chemical mixing state in Riverside, CA, Atmos. Environ., 59 (2012), 587-596. doi: 10.1016/j.atmosenv.2012.05.032.
[38]	N. Riemer, H. Vogel, B. Vogel and F. Fiedler, Modeling aerosols on the mesoscale $\gamma$, part Ⅰ: Treatment of soot aerosol and its radiative effects, J. Geophys. Res., 108 (2003), 4601. doi: 10.1029/2003JD003448.
[39]	N. Riemer, M. West, R. A. Zaveri and R. C. Easter, Simulating the evolution of soot mixing state with a particle-resolved aerosol model, J. Geophys. Res., (2009), D09202. doi: 10.1029/2008JD011073.
[40]	J. H. Seinfeld and S. Pandis, Atmospheric Chemistry and Physics, Wiley, 2016.
[41]	S. Shima, K. Kusano, A. Kawano, T. Sugiyama and S. Kawahara, The super-droplet method for the numerical simulation of clouds and precipitation: A particle-based and probabilistic microphysics model coupled with a non-hydrostatic model, Q. J. R. Meteorol. Soc., 135 (2009), 1307-1320. doi: 10.1002/qj.441.
[42]	A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Chapman & Hall, London, 1995.
[43]	J. Tian, N. Riemer, M. West, L. Pfaffenberger, H. Schlager and A. Petzold, Modeling the evolution of aerosol particles in a ship plume using PartMC-MOSAIC, Atmos. Chem. Phys., 14 (2014), 5327-5347. doi: 10.5194/acp-14-5327-2014.
[44]	C. G. Wells and M. Kraft, Direct simulation and mass flow stochastic algorithms to solve a sintering-coagulation equation, Monte Carlo Methods Appl., 11 (2005), 175-197. doi: 10.1515/156939605777585980.
[45]	M. West, N. Riemer, J. Curtis, M. Michelotti and J. Tian, PartMC: Particle-resolved Monte-Carlo atmospheric aerosol simulation, version 2.5.0, 2018. doi: 10.5281/zenodo.1490925.
[46]	C. Yoon and R. McGraw, Representation of generally mixed multivariate aerosols by the quadrature method of moments: Ⅱ. Aerosol dynamics, J. Aerosol Sci., 35 (2004), 577-598. doi: 10.1016/j.jaerosci.2003.11.012.
[47]	H. Zhao and C. Zheng, Correcting the Multi-Monte Carlo Method for particle coagulation, Powder Technol., 193 (2009), 120-123. doi: 10.1016/j.powtec.2009.01.019.
[48]	H. Zhao, C. Zheng and M. Xu, Multi-Monte Carlo Method for coagulation and condensation/evaporation in dispersed systems, J. Colloid Interface Sci., 286 (2005), 195-208. doi: 10.1016/j.jcis.2004.12.037.
[49]	H. Zhao, F. E. Kruis and C. Zheng, Reducing statistical noise and extending the size spectrum by applying weighted simulation particles in Monte Carlo simulation of coagulation, Aerosol Sci. Technol., 43 (2009), 781-793. doi: 10.1080/02786820902939708.

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

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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

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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} $

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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

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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

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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)

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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 $)

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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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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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American Institute of Mathematical Sciences

Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation

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Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation

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