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Mori-Zwanzig reduced models for uncertainty quantification
Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation
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 |
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.
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. |







Variables | Meaning | Range |
physical particle class | ||
event outcomes | ||
basis vector | ||
physical particle size | ||
coagulation kernel | ||
event rate | ||
physical number concentration | ||
total number of computational particles | ||
probability | ||
number of computational particles | ||
event rate | ||
selection rate function | ||
computational volume | ||
weighting function | ||
physical particle concentration | ||
computational particle concentration | ||
event jump |
Variables | Meaning | Range |
physical particle class | ||
event outcomes | ||
basis vector | ||
physical particle size | ||
coagulation kernel | ||
event rate | ||
physical number concentration | ||
total number of computational particles | ||
probability | ||
number of computational particles | ||
event rate | ||
selection rate function | ||
computational volume | ||
weighting function | ||
physical particle concentration | ||
computational particle concentration | ||
event jump |
[1] |
Philippe Laurençot, Barbara Niethammer, Juan J.L. Velázquez. Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel. Kinetic and Related Models, 2018, 11 (4) : 933-952. doi: 10.3934/krm.2018037 |
[2] |
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