Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method

Hatim Tayeq; Amal Bergam; Anouar El Harrak; Kenza Khomsi

doi:10.3934/dcdss.2020400

Previous Article
Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives
DCDS-S Home
This Issue
Next Article
Optimal control strategy for an age-structured SIR endemic model

July 2021, 14(7): 2557-2570. doi: 10.3934/dcdss.2020400

Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method

Hatim Tayeq ^1,, , Amal Bergam ^1, , Anouar El Harrak ^1, and Kenza Khomsi ^2,

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

^* Corresponding author: tayeq.hatim@gmail.com (Hatim Tayeq)

Received July 2019 Revised October 2019 Published July 2021 Early access June 2020

Fund Project: This work was partially supported by the project PICS Mairoc with a corporation with the Jean Leray Laboratory of Nantes university (France) and the National Meteorological Direction (DMN) of 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).

Keywords: Self-adaptive algorithm, a posteriori error estimators, finite volume method, pollutant transport, air quality forecast, parametrizations.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Hatim Tayeq, Amal Bergam, Anouar El Harrak, Kenza Khomsi. Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400

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/.
[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/.
[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.

Figure 1. Vertex-centred cell in 2D

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The measurement stations on the Grand Casablanca area

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Numerical simulations obtained using a uniform mesh with 443 elements at two times $ t = 50 $, (A), and $ t = 1400 $, (B)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Numerical simulations obtained using a uniform mesh with 22887 elements at two times $ t = 50 $, (A), and $ t = 1400 $, (B)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. (A) and (B) are the approximate solution using a adaptive mesh at $ t = 50 $ and $ t = 1400 $ respectively in level 2

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. (A), (B) and (C) are the adaptively refined meshes at $ t = 50 $ in level 1, 2 and 3 respectively

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. (A), (B) and (C) are the adaptively refined meshes at $ t = 1400 $ in level 1, 2 and 3 respectively

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Forecasted concentration of ozone before and after the implementation of our algorithm in two positions: Jahid station, (A) and Khansae station, (B)

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. The calculated parameters for the numerical model

Parameter	Signification	Value
$ D $	Diffusion coefficient	$ 95.3\ m^2.s^{-1} $
$ V $	Velocity vector	$ [3.9\ m.s^{-1}, 1.58\ m.s^{-1}] $
$ r $	Reaction balance	$ 3.9 $
$ f $	Second member	$ 2.95\ \mu g.m^{-3} $
$ c_0 $	Initial concentration	$ 64\ \mu g.m^{-3} $

Table Options

Download as excel

Parameter	Signification	Value
$ D $	Diffusion coefficient	$ 95.3\ m^2.s^{-1} $
$ V $	Velocity vector	$ [3.9\ m.s^{-1}, 1.58\ m.s^{-1}] $
$ r $	Reaction balance	$ 3.9 $
$ f $	Second member	$ 2.95\ \mu g.m^{-3} $
$ c_0 $	Initial concentration	$ 64\ \mu g.m^{-3} $

Table 2. The self-adaptive algorithm at an iteration

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.

Table Options

Download as excel

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.

Table 3. The results obtained before and after the application of the self-adaptive algorithm of the mesh at $ t = 50 $

	Level	Number of	jump	residual	CPU
		elements	indicator	indicator	time
primal mesh		$ 443 $	$ 5.0132e-03 $	$ 1.6088e-04 $	$ 1.4053\ s $
$ t=50 $	$ 1 $	$ 196 $	$ 9.8885e-05 $	$ 9.4178e-04 $	$ 6.0905\ s $
	$ 2 $	$ 278 $	$ 3.1335e-05 $	$ 2.6348e-04 $	$ 7.2734\ s $
	$ 3 $	$ 629 $	$ 7.8928e-06 $	$ 3.1045e-06 $	$ 8.2617\ s $
uniform mesh		$ 22887 $	$ 1.3548e-07 $	$ 1.9964e-06 $	$ 53.4408\ s $

Table Options

Download as excel

	Level	Number of	jump	residual	CPU
		elements	indicator	indicator	time
primal mesh		$ 443 $	$ 5.0132e-03 $	$ 1.6088e-04 $	$ 1.4053\ s $
$ t=50 $	$ 1 $	$ 196 $	$ 9.8885e-05 $	$ 9.4178e-04 $	$ 6.0905\ s $
	$ 2 $	$ 278 $	$ 3.1335e-05 $	$ 2.6348e-04 $	$ 7.2734\ s $
	$ 3 $	$ 629 $	$ 7.8928e-06 $	$ 3.1045e-06 $	$ 8.2617\ s $
uniform mesh		$ 22887 $	$ 1.3548e-07 $	$ 1.9964e-06 $	$ 53.4408\ s $

Table 4. The results obtained before and after the application of the self-adaptive algorithm of the mesh at $ t = 1400 $

	Level	Number of	Jump	Residual	CPU
		elements	indicator	indicator	time
Primal mesh		$ 443 $	$ 1.6483e-05 $	$ 1.9601e-05 $	$ 1.4053\ s $
Adaptive mesh	$ 1 $	$ 828 $	$ 1.3212e-07 $	$ 1.9516e-06 $	$ 18.1240\ s $
	$ 2 $	$ 1374 $	$ 7.2050e-08 $	$ 1.5499e-07 $	$ 25.0560\ s $
uniform mesh		$ 22887 $	$ 1.1405e-08 $	$ 1.2477e-07 $	$ 95.6868\ s $

Table Options

Download as excel

	Level	Number of	Jump	Residual	CPU
		elements	indicator	indicator	time
Primal mesh		$ 443 $	$ 1.6483e-05 $	$ 1.9601e-05 $	$ 1.4053\ s $
Adaptive mesh	$ 1 $	$ 828 $	$ 1.3212e-07 $	$ 1.9516e-06 $	$ 18.1240\ s $
	$ 2 $	$ 1374 $	$ 7.2050e-08 $	$ 1.5499e-07 $	$ 25.0560\ s $
uniform mesh		$ 22887 $	$ 1.1405e-08 $	$ 1.2477e-07 $	$ 95.6868\ s $

[1]	Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459
[2]	Ya-Zheng Dang, Zhong-Hui Xue, Yan Gao, Jun-Xiang Li. Fast self-adaptive regularization iterative algorithm for solving split feasibility problem. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1555-1569. doi: 10.3934/jimo.2019017
[3]	Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial and Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255
[4]	Yuan Shen, Chang Liu, Yannian Zuo, Xingying Zhang. A modified self-adaptive dual ascent method with relaxed stepsize condition for linearly constrained quadratic convex optimization. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022101
[5]	Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062
[6]	Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[7]	Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[8]	Ruiqi Li, Yifan Chen, Xiang Zhao, Yanli Hu, Weidong Xiao. Time series based urban air quality predication. Big Data & Information Analytics, 2016, 1 (2&3) : 171-183. doi: 10.3934/bdia.2016003
[9]	Hsueh-Chen Lee, Hyesuk Lee. An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows. Electronic Research Archive, 2021, 29 (4) : 2755-2770. doi: 10.3934/era.2021012
[10]	John D. Towers. An explicit finite volume algorithm for vanishing viscosity solutions on a network. Networks and Heterogeneous Media, 2022, 17 (1) : 1-13. doi: 10.3934/nhm.2021021
[11]	Lili Ju, Wensong Wu, Weidong Zhao. Adaptive finite volume methods for steady convection-diffusion equations with mesh optimization. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 669-690. doi: 10.3934/dcdsb.2009.11.669
[12]	Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051
[13]	Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial and Management Optimization, 2022, 18 (1) : 239-265. doi: 10.3934/jimo.2020152
[14]	Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055
[15]	Walter Allegretto, Yanping Lin, Ningning Yan. A posteriori error analysis for FEM of American options. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 957-978. doi: 10.3934/dcdsb.2006.6.957
[16]	Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[17]	Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[18]	Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
[19]	Tan Bui-Thanh, Omar Ghattas. A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors. Inverse Problems and Imaging, 2015, 9 (1) : 27-53. doi: 10.3934/ipi.2015.9.27
[20]	Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

2020 Impact Factor: 2.425

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 2.425

Tools

Figures and Tables

2020 Impact Factor: 2.425

American Institute of Mathematical Sciences

Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors