Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-28T18:55:10.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical Modeling of Atmospheric Flow and Computation of Convex Envelopes

Published online by Cambridge University Press:  10 August 2011

A. Caboussat
A. Caboussat*
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3008, USA
*
⋆ Corresponding author. E-mail: caboussat@math.uh.edu
Get access

Abstract

Atmospheric flow equations govern the time evolution of chemical concentrations in the atmosphere. When considering gas and particle phases, the underlying partial differential equations involve advection and diffusion operators, coagulation effects, and evaporation and condensation phenomena between the aerosol particles and the gas phase. Operator splitting techniques are generally used in global air quality models. When considering organic aerosol particles, the modeling of the thermodynamic equilibrium of each particle leads to the determination of the convex envelope of the energy function. Two strategies are proposed to address the computation of convex envelopes. The first one is based on a primal-dual interior-point method, while the second one relies on a variational formulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finite element techniques. Numerical experiments are presented for chemical systems of atmospheric interest, in order to compute convex envelopes in various space dimensions.

Keywords

atmospheric flowair qualityconvex envelopesinterior-point methodvariational problemaugmented Lagrangian
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 44 - 66
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Amundson, N. R., Caboussat, A., He, J. W., Landry, C., Seinfeld, J. H.. A dynamic optimization problem related to organic aerosols. C. R. Acad. Sci., 344 (2007), No. 8, 519–522. CrossRefGoogle Scholar
Amundson, N. R., Caboussat, A., He, J. W., Landry, C., Tong, C., Seinfeld, J. H.. A new atmospheric aerosol phase equilibrium model (UHAERO): organic systems. Atmos. Chem. Phys., 7 (2007), 4675–4698. CrossRefGoogle Scholar
Amundson, N. R., Caboussat, A., He, J. W., Martynenko, A. V., Seinfeld, J. H., Yoo, K. Y.. A new inorganic atmospheric aerosol phase equilibrium model (UHAERO). Atmos. Chem. Phys., 6 (2006), 975–992. CrossRefGoogle Scholar
Amundson, N. R., Caboussat, A., He, J. W., Seinfeld, J. H.. Primal-dual interior-point algorithm for chemical equilibrium problems related to modeling of atmospheric organic aerosols. J. Optim. Theory Appl., 130 (2006), No. 3, 375–407. CrossRefGoogle Scholar
Benson, H. Y., Shanno, D. F.. Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts. Comput. Optim. Appl., 40 (2008), No. 2, 143–189. CrossRefGoogle Scholar
Caboussat, A.. Primal-dual interior-point method for thermodynamic gas-particle partitioning. Computational Optimization and Applications, 48 (2011), No. 3, 717–745. CrossRefGoogle Scholar
Caboussat, A., Glowinski, R.. A numerical method for a non-smooth advection-diffusion problem arising in sand mechanics. Com. Pure. Appl. Anal, 8 (2008), No. 1, 161–178. CrossRefGoogle Scholar
Caboussat, A., Glowinski, R., Pons, V.. An augmented Lagrangian approach to the numerical solution of a non-smooth eigenvalue problem. J. Numer. Math, 17 (2009), No. 1, 3–26. CrossRefGoogle Scholar
Caboussat, A., Landry, C., Rappaz, J.. Optimization problem coupled with differential equations: A numerical algorithm mixing an interior-point method and event detection. J. Optim. Theory Appl., 147 (2010), No. 1, 141–156. CrossRefGoogle Scholar
Carmichael, G. R., Peters, L. K., Kitada, T.. A second generation model for the regional-scale transport/chemistry/deposition. Atm. Env., 20 (1986), 173. CrossRefGoogle Scholar
Dean, E. J., Glowinski, R.. An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions. Electronic Transactions in Numerical Analysis, 22 (2006), 71–96. Google Scholar
Dean, E. J., Glowinski, R., Guidoboni, G.. On the numerical simulation of Bingham visco-plastic flow: old and new results. Journal of Non Newtonian Fluid Mechanics, 142 (2007), 36–62. CrossRefGoogle Scholar
A. V. Fiacco, G. P. McCormick. Nonlinear programming : sequential unconstrained minimization techniques, Wiley, New York, 1968.
M. Fortin, R. Glowinski. Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Studies in Mathematics and Its Applications. Elsevier Science Ltd, 1983.
R. Glowinski. Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, NY, 1984.
R. Glowinski, P. Le Tallec. Augmented Lagrangians and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.
Gondzio, J., Grothey, A.. A new unblocking technique to warmstart interior point methods based on sensitivity analysis. SIAM Journal on Optimization, 19 (2008), No. 3, 1184–1210. CrossRefGoogle Scholar
M. Z. Jacobson. Fundamentals of Atmospheric Modeling, Cambridge, second edition, 2005.
C. Landry. Numerical Analysis of Optimization-Constrained Differential Equations: Applications to Atmospheric Chemistry. PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2009. Available at http://library.epfl.ch/theses/?nr=4345.
Landry, C., Caboussat, A., Hairer, E.. Solving optimization-constrained differential equations with discontinuity points, with application to atmospheric chemistry. SIAM J. Sci. Comp., 31 (2009), No. 5, 3806–3826. CrossRefGoogle Scholar
Lanser, D., Verwer, J.. Analysis of operator splitting for advection-diffusion-reaction problems from air pollution modelling. J. Comput. Appl. Math., 111 (1999), 201–216. CrossRefGoogle Scholar
McDonald, C. M., Floudas, C. A.. GLOPEQ: A new computational tool for the phase and chemical equilibrium problem. Computers and Chemical Engineering, 21 (1996), No. 1, 1–23. CrossRefGoogle Scholar
McRae, G. J., Goodin, W. R., Seinfeld, J. H.. Numerical solution of the atmospheric diffusion equation for chemically reacting flows. J. Comput. Phys., 45 (1982), No. 1, 1–42. CrossRefGoogle Scholar
Meng, Z., Dabdub, D., Seinfeld, J. H.. Size-resolved and chemically resolved model of atmospheric aerosol dynamics. J. Geophys. Res., 103 (1998), 3419–3436. CrossRefGoogle Scholar
Nguyen, K., Caboussat, A., Dabdub, D.. Mass conservative, positive definite integrator for atmospheric chemical dynamics. Atmos. Env., 43 (2009), No. 40, 6287–6295. CrossRefGoogle Scholar
Nguyen, K., Dabdub, D.. Semi-lagrangian flux scheme for the solution of the aerosol condensation/evaporation equation. Aerosol Science & Technology, 36 (2002), 407–418. CrossRefGoogle Scholar
R. T. Rockafellar. Convex analysis, Princeton University Press, Princeton, NJ, 1970.
J. H. Seinfeld, S. N. Pandis. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Wiley, New York, 1998.
S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, H.L. Miller, editors. Intergovernmental Panel on Climate Change: Fourth Assessment Report: Climate Change 2007, The Physical Science Basis, Cambridge University Press, 2007.
Sportisse, B.. A review of current issues in air pollution modeling and simulation. Comput. Geosci., 11 (2007), 159–181. CrossRefGoogle Scholar
Verwer, J.G., Hundsdorfer, W., Blom, J.G.. Numerical time integration for air pollution models. Surveys Math. Ind., 10 (2002), 107–174. Google Scholar
Zaveri, R. A., Easter, R. C., Fast, J. D., Peters, L. K.. Model for simulating aerosol interactions and chemistry (MOSAIC). J. Geophys. Res. D (Atmospheres), 113 (2008), No. D13, D13204. CrossRefGoogle Scholar
Zaveri, R. A., Easter, R. C., Peters, L. K.. A computationally efficient multicomponent equilibrium solver for aerosols (MESA). J. Geophys. Res. D (Atmospheres), 110 (2005), No. D24, D24203. CrossRefGoogle Scholar