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

Article Contents

2021, Volume 14, Issue 7: 2557-2570. Doi: 10.3934/dcdss.2020400

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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

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

Received: June 30, 2019

Revised: September 30, 2019

Early access: June 2020

Published: July 2021

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

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Abstract

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).

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

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Figure 1. Vertex-centred cell in 2D

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Figure 2. The measurement stations on the Grand Casablanca area

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

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

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

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

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

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

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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} $

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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.

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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 $

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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 $

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