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13 - A quasi-Monte Carlo method for the coagulation equation

Published online by Cambridge University Press:  18 December 2014

Christian Lécot
Affiliation:
Université de Savoie, Le Bourget-du-Lac
Ali Tarhini
Affiliation:
Université Libanaise, Nabatieh
Gerhard Larcher
Affiliation:
Johannes Kepler Universität Linz
Friedrich Pillichshammer
Affiliation:
Johannes Kepler Universität Linz
Arne Winterhof
Affiliation:
Austrian Academy of Sciences, Linz
Chaoping Xing
Affiliation:
Nanyang Technological University, Singapore
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Publisher: Cambridge University Press
Print publication year: 2014

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References

[1] D. J., Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48, 1999.Google Scholar
[2] H., Babovsky, On a Monte Carlo scheme for Smoluchowski's coagulation equation. Monte Carlo Methods Appl. 5, 1–18, 1999.Google Scholar
[3] J., Dick and F., Pillichshammer, Digital Nets and Sequences. Cambridge University Press, Cambridge, 2010.
[4] R. B., Diemer and J. H., Olson, A moment methodology for coagulation and breakage problems: part 2 – moment models and distribution reconstruction. Chem. Eng. Sci. 57, 2211–2228, 2002.Google Scholar
[5] R. L., Drake, A general mathematical survey of the coagulation equation. In: G. M., Hidy and J. R., Brock (eds.), Topics in Current Aerosol Research, Part 2, pp. 201–376. Pergamon Press, Oxford, 1972.
[6] M., Drmota and R. F., Tichy, Sequences, Discrepancies and Applications. Springer-Verlag, Berlin, 1997.
[7] P. B., Dubovskiῐ, Mathematical Theory of Coagulation. Lecture Notes 23. Global Analysis Research Center, Seoul National University, 1994.
[8] A., Eibeck and W., Wagner, An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena. SIAM J. Sci. Comput. 22, 802–821, 2000.Google Scholar
[9] A., Eibeck and W., Wagner, Stochastic particle approximations for Smoluchowski's coagulation equation. Ann. Appl. Prob. 11, 1137–1165, 2001.Google Scholar
[10] F., Filbet and P., Laurençot, Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25, 2004–2028, 2004.Google Scholar
[11] D. T., Gillespie, An exact method for numerically simulating the stochastic coalescence process in a cloud. J. Atmos. Sci. 32, 1977–1989, 1975.Google Scholar
[12] I. S., Gradshteyn and I. M., Ryzhik, Table of Integrals, Series and Products, seventh edition. Academic Press, San Diego, CA, 2007.
[13] M., Kostoglou, Extended cell average technique for the solution of coagulation equation. J. Colloid Interface Sci. 306, 72–81, 2007.Google Scholar
[14] L., Kuipers and H., Niederreiter, Uniform Distribution of Sequences. John Wiley & Sons, New York, 1974.
[15] S., Kumar and D., Ramkrishna, On the solution of population balance equation by discretization – I. A fixed pivot technique. Chem. Eng. Sci. 51, 1311–1332, 1996.Google Scholar
[16] J., Kumar, M., Peglow, G., Warnecke, S., Heinrich and L., Mörl, Improved accuracy and convergence of discretized population balance for aggregation: the cell average technique. Chem.Eng.Sci. 61, 3327–3342, 2006.Google Scholar
[17] J., Kumar, G., Warnecke, M., Peglow and S., Heinrich, Comparison of numerical methods for solving population balance equations incorporating aggregation and breakage. Powder Technol. 189, 218–229, 2009.Google Scholar
[18] G., Larcher, Digital point sets: analysis and application. In: P., Hellekalek and G., Larcher (eds.), Random and Quasi-Random Point Sets. Lecture Notes in Statistics, volume 138, pp. 167–222. Springer-Verlag, Berlin, 1998.
[19] C., Lécot, A direct simulation Monte Carlo scheme and uniformly distributed sequences for solving the Boltzmann equation. Computing 41, 41–57, 1989.Google Scholar
[20] C., Lécot, A quasi-Monte Carlo method for the Boltzmann equation. Math. Comput. 56, 621–644, 1991.Google Scholar
[21] C., Lécot, Error bounds for quasi-Monte Carlo integration with nets. Math. Comput. 65, 179–187, 1996.Google Scholar
[22] C., Lécot and A., Tarhini, A quasi-stochastic simulation of the general dynamics equation for aerosols. Monte Carlo Methods Appl. 13, 369–388, 2007.Google Scholar
[23] C., Lécot and W., Wagner, A quasi-Monte Carlo scheme for Smoluchowski's coagulation equation. Math. Comput. 73, 1953–1966, 2004.Google Scholar
[24] C., Lécot, M., Tembely, A., Soucemarianadin and A., Tarhini, Numerical simulation of the drop size distribution in a spray. In: L., Plaskota and H., Wozniakowski (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 503–517. Springer-Verlag, Berlin, 2012.
[25] G., Madras and B. J., McCoy, Numerical and similarity solutions for reversible population balance equations with size-dependent rates. J. Colloid Interface Sci. 246, 356–365, 2002.Google Scholar
[26] Z. A., Melzak, The effect of coalescence in certain collision processes. Q. J. Appl. Math. 11, 231–234, 1953.Google Scholar
[27] Z. A., Melzak, A scalar transport equation. Trans. Am. Math. Soc. 85, 547–560, 1957.Google Scholar
[28] W. J., Morokoff and R. E., Caflisch, Quasi-Monte Carlo integration. J. Comput. Phys. 122, 218–230, 1995.Google Scholar
[29] H., Müller, Zur allgemeinen Theorie der raschen Koagulation. Kolloidchem. Beih. 27, 223–250, 1928.Google Scholar
[30] H., Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Am. Math. Soc. 84, 957–1041, 1978.Google Scholar
[31] H., Niederreiter, Point sets and sequences with small discrepancy. Monatsh. Math. 104, 273–337, 1987.Google Scholar
[32] H., Niederreiter, Low-discrepancy and low-dispersion sequences. J. Number Theory 30, 51–70, 1988.Google Scholar
[33] H., Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, PA, 1992.
[34] H., Niederreiter, Constructions of (t, m, s)-nets. In: H., Niederreiter and J., Spanier (eds.), Monte Carlo and Quasi-Monte Carlo Methods 1998, pp. 70–85. Springer-Verlag, Berlin, 2000.
[35] H., Niederreiter, Constructions of (t, m, s)-nets and (t, s)-sequences. Finite Fields Appl. 11, 578–600, 2005.Google Scholar
[36] T. E., Ramabhadran, T. W., Peterson and J. H., Seinfeld, Dynamics of aerosol coagulation and condensation. AIChE J. 22, 840–851, 1976.Google Scholar
[37] T. E. W., Schumann, Theoretical aspects of the size distribution of fog particles. Q. J. R. Meteorol. Soc. 66, 195–207, 1940.Google Scholar
[38] W. T., Scott, Analytic studies of cloud droplet coalescence I. J. Atmos. Sci. 25, 54–65, 1968.Google Scholar
[39] M., Smith and T., Matsoukas, Constant-number Monte Carlo simulation of population balances. Chem. Eng. Sci. 53, 1777–1786, 1998.Google Scholar
[40] M. von, Smoluchowski, Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 92, 129–168, 1916.Google Scholar
[41] M. von, Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen. Phys. Z. 17, 557–599, 1916.Google Scholar
[42] M., Sommer, F., Stenger, W., Peukert and N. J., Wagner, Agglomeration and breakage of nanoparticles in stirred media mills – a comparison of different methods and models. Chem. Eng. Sci. 61, 135–148, 2006.Google Scholar
[43] J., Wei and F. E., Kruis, GPU-accelerated Monte Carlo simulation of particle coagulation based on the inverse method. J. Comput. Phys. 249, 67–79, 2013.Google Scholar
[44] S. K., Zaremba, Some applications of multidimensional integration by parts. Ann. Pol. Math. 21, 85–96, 1968.Google Scholar
[45] H., Zhao and C., Zheng, A new event-driven constant-volume method for solution of the time evolution of particle size distribution. J. Comput. Phys. 228, 1412–1428, 2009.Google Scholar
[46] H., Zhao and C., Zheng, Correcting the multi-Monte Carlo method for particle coagulation. Powder Technol. 193, 120–123, 2009.Google Scholar
[47] H., Zhao, C., Zheng and M., Xu, Multi-Monte Carlo method for particle coagulation: description and validation. Appl. Math. Comput. 167, 1383–1399, 2005.Google Scholar
[48] H., Zhao, A., Maisels, T., Matsoukas and C., Zheng, Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems. Powder Technol. 173, 38–50, 2007.Google Scholar
[49] Y., Zou, M. E., Kavousanakis, I. G., Kevrekidis and R. O., Fox, Coarse-grained computation for particle coagulation and sintering processes by linking quadrature method of moments with Monte Carlo. J. Comput. Phys. 229, 5299–5314, 2010.Google Scholar

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