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Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme
Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method
1. | MAE2D laboratory, University Abdelmalek Essaadi, Polydisciplinary Faculty of Larache, Road of Rabat, Larache, Morocco |
2. | National Meteorological Direction of Morocco, Airport Casa-Anfa, Casa Oasis - Casablanca, Morocco |
In this work, we present a self-adaptive algorithm based on the techniques of the a posteriori estimates for the transport equation modeling the dispersion of pollutants in the atmospheric boundary layer at the local scale (industrial accident, urban air quality). The goal is to provide a powerful model for forecasting pollutants concentrations with better manipulation of available computing resources.
This analysis is based on a vertex-centered Finite Volume Method in space and an implicit Euler scheme in time. We apply and validate our model, using a self-adaptive algorithm, with real atmospheric data of the Grand Casablanca area (Morocco).
References:
[1] |
M. Afif, A. Bergam, Z. Mghazli and R. Verfurth,
A posteriori estimators of the finite volume discretization of an elliptic problem, Numer. Algorithms, 34 (2003), 127-136.
doi: 10.1023/B:NUMA.0000005400.45852.f3. |
[2] |
B. Amaziane, A. Bergam, M. El Ossmani and Z. Mghazli,
A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, Internat. J. Numer. Methods Fluids, 59 (2009), 259-284.
doi: 10.1002/fld.1456. |
[3] |
A. Bergam, C. Bernardi, F. Hecht and Z. Mghazli,
Error indicators for the mortar finite element discretization of a parabolic problem, Numer. Algorithms, 34 (2003), 187-201.
doi: 10.1023/B:NUMA.0000005362.95126.4c. |
[4] |
A. Bergam, Z. Mghazli and R. Verfürth,
Estimations a posteriori d'un schéma de volumes finis pour un problème non linéaire, Numer. Math., 95 (2003), 599-624.
doi: 10.1007/s00211-003-0460-2. |
[5] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[6] |
W. Gong and H.-R. Cho,
A numerical scheme for the integration of the gas-phase chemical rate equations in 3d atmospheric models, Atmospheric Environment, 27 (1993), 2147-2160.
doi: 10.1016/0960-1686(93)90044-Y. |
[7] |
J.-F. Louis,
A parametric model of vertical eddy fluxes in the atmosphere, Boundary-Layer Meteorology, 17 (1979), 187-202.
doi: 10.1007/BF00117978. |
[8] |
V. Mallet, A. Pourchet, D. Quélo and B. Sportisse,
Investigation of some numerical issues in a chemistry-transport model: Gas-phase simulations, J. Geophysical Research, 112 (2007), 301-317.
doi: 10.1029/2006JD008373. |
[9] |
V. Mallet and B. Sportisse,
3-D chemistry-transport model Polair: Numerical issues, validation and automatic-differentiation strategy, Atmos. Chem. Phys. Discuss., 4 (2004), 1371-1392.
doi: 10.5194/acpd-4-1371-2004. |
[10] |
V. Mallet and B. Sportisse,
Uncertainty in a chemistry-transport model due to physical parameterizations and numerical approximations: An ensemble approach applied to ozone modeling, J. Geophys. Res., 111 (2006), 1-15.
doi: 10.1029/2005JD006149. |
[11] |
W. R. Stockwell, F. Kirchner, M. Kuhn and S. Seefeld, A new mechanism for regional atmospheric chemistry modeling, J. Geophys. Res., 102 (1997).
doi: 10.1029/97JD00849. |
[12] |
W. R. Stockwell, P. Middleton, J. S. Chang and X. Tang,
The second generation regional acid deposition model chemical mechanism for regional air quality modeling, J. Geophys. Res., 95 (1990), 343-367.
doi: 10.1029/JD095iD10p16343. |
[13] |
R. B. Stull, An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library, 13, Springer, Dordrecht, 1998.
doi: 10.1007/978-94-009-3027-8. |
[14] |
World Helth Organization, Ambient Air Pollution: Pollutants, WHO Report, WHO Library Cataloguing-in-Publication Data, 2018. Available from: http://www.who.int/airpollution/ambient/pollutants/en/. Google Scholar |
[15] |
L. Wu, V. Mallet, M. Bocquet and B. Sportisse,
A comparison study of data assimilation algorithms for ozone forecasts, J. Geophys. Res., 113 (2008), 1-17.
doi: 10.1029/2008JD009991. |
show all references
References:
[1] |
M. Afif, A. Bergam, Z. Mghazli and R. Verfurth,
A posteriori estimators of the finite volume discretization of an elliptic problem, Numer. Algorithms, 34 (2003), 127-136.
doi: 10.1023/B:NUMA.0000005400.45852.f3. |
[2] |
B. Amaziane, A. Bergam, M. El Ossmani and Z. Mghazli,
A posteriori estimators for vertex centred finite volume discretization of a convection-diffusion-reaction equation arising in flow in porous media, Internat. J. Numer. Methods Fluids, 59 (2009), 259-284.
doi: 10.1002/fld.1456. |
[3] |
A. Bergam, C. Bernardi, F. Hecht and Z. Mghazli,
Error indicators for the mortar finite element discretization of a parabolic problem, Numer. Algorithms, 34 (2003), 187-201.
doi: 10.1023/B:NUMA.0000005362.95126.4c. |
[4] |
A. Bergam, Z. Mghazli and R. Verfürth,
Estimations a posteriori d'un schéma de volumes finis pour un problème non linéaire, Numer. Math., 95 (2003), 599-624.
doi: 10.1007/s00211-003-0460-2. |
[5] |
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58090-1. |
[6] |
W. Gong and H.-R. Cho,
A numerical scheme for the integration of the gas-phase chemical rate equations in 3d atmospheric models, Atmospheric Environment, 27 (1993), 2147-2160.
doi: 10.1016/0960-1686(93)90044-Y. |
[7] |
J.-F. Louis,
A parametric model of vertical eddy fluxes in the atmosphere, Boundary-Layer Meteorology, 17 (1979), 187-202.
doi: 10.1007/BF00117978. |
[8] |
V. Mallet, A. Pourchet, D. Quélo and B. Sportisse,
Investigation of some numerical issues in a chemistry-transport model: Gas-phase simulations, J. Geophysical Research, 112 (2007), 301-317.
doi: 10.1029/2006JD008373. |
[9] |
V. Mallet and B. Sportisse,
3-D chemistry-transport model Polair: Numerical issues, validation and automatic-differentiation strategy, Atmos. Chem. Phys. Discuss., 4 (2004), 1371-1392.
doi: 10.5194/acpd-4-1371-2004. |
[10] |
V. Mallet and B. Sportisse,
Uncertainty in a chemistry-transport model due to physical parameterizations and numerical approximations: An ensemble approach applied to ozone modeling, J. Geophys. Res., 111 (2006), 1-15.
doi: 10.1029/2005JD006149. |
[11] |
W. R. Stockwell, F. Kirchner, M. Kuhn and S. Seefeld, A new mechanism for regional atmospheric chemistry modeling, J. Geophys. Res., 102 (1997).
doi: 10.1029/97JD00849. |
[12] |
W. R. Stockwell, P. Middleton, J. S. Chang and X. Tang,
The second generation regional acid deposition model chemical mechanism for regional air quality modeling, J. Geophys. Res., 95 (1990), 343-367.
doi: 10.1029/JD095iD10p16343. |
[13] |
R. B. Stull, An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library, 13, Springer, Dordrecht, 1998.
doi: 10.1007/978-94-009-3027-8. |
[14] |
World Helth Organization, Ambient Air Pollution: Pollutants, WHO Report, WHO Library Cataloguing-in-Publication Data, 2018. Available from: http://www.who.int/airpollution/ambient/pollutants/en/. Google Scholar |
[15] |
L. Wu, V. Mallet, M. Bocquet and B. Sportisse,
A comparison study of data assimilation algorithms for ozone forecasts, J. Geophys. Res., 113 (2008), 1-17.
doi: 10.1029/2008JD009991. |








Parameter | Signification | Value |
Diffusion coefficient | ||
Velocity vector | ||
Reaction balance | ||
Second member | ||
Initial concentration |
Parameter | Signification | Value |
Diffusion coefficient | ||
Velocity vector | ||
Reaction balance | ||
Second member | ||
Initial concentration |
Step 1: | Mesh generating and data input. |
Step 2: | Solve the discrete problem. |
Step 3: | Calculate the local error indicators. |
Step 4: | Mark mesh cells and adapt mesh. |
Step 5: | Solve the discrete problem in the new adapted mesh. |
Step 6: | if the stopping test is satisfied, go to the step 7 |
else, go to the step 3. | |
Step 7: | Interpolate solution and visualization. |
Step 1: | Mesh generating and data input. |
Step 2: | Solve the discrete problem. |
Step 3: | Calculate the local error indicators. |
Step 4: | Mark mesh cells and adapt mesh. |
Step 5: | Solve the discrete problem in the new adapted mesh. |
Step 6: | if the stopping test is satisfied, go to the step 7 |
else, go to the step 3. | |
Step 7: | Interpolate solution and visualization. |
Level | Number of | jump | residual | CPU | |
elements | indicator | indicator | time | ||
primal mesh | |||||
uniform mesh |
Level | Number of | jump | residual | CPU | |
elements | indicator | indicator | time | ||
primal mesh | |||||
uniform mesh |
Level | Number of | Jump | Residual | CPU | |
elements | indicator | indicator | time | ||
Primal mesh | |||||
Adaptive mesh | |||||
uniform mesh |
Level | Number of | Jump | Residual | CPU | |
elements | indicator | indicator | time | ||
Primal mesh | |||||
Adaptive mesh | |||||
uniform mesh |
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