    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

 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).

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 & Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400
##### References:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  J.-F. Louis, A parametric model of vertical eddy fluxes in the atmosphere, Boundary-Layer Meteorology, 17 (1979), 187-202.  doi: 10.1007/BF00117978. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  R. B. Stull, An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library, 13, Springer, Dordrecht, 1998. doi: 10.1007/978-94-009-3027-8. Google Scholar  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  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. Google Scholar Numerical simulations obtained using a uniform mesh with 443 elements at two times $t = 50$, (A), and $t = 1400$, (B) Numerical simulations obtained using a uniform mesh with 22887 elements at two times $t = 50$, (A), and $t = 1400$, (B) (A) and (B) are the approximate solution using a adaptive mesh at $t = 50$ and $t = 1400$ respectively in level 2 (A), (B) and (C) are the adaptively refined meshes at $t = 50$ in level 1, 2 and 3 respectively (A), (B) and (C) are the adaptively refined meshes at $t = 1400$ in level 1, 2 and 3 respectively Forecasted concentration of ozone before and after the implementation of our algorithm in two positions: Jahid station, (A) and Khansae station, (B)
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 $64\ \mu g.m^{-3}$
 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 $64\ \mu g.m^{-3}$
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.
 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.
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 $22887$ $1.3548e-07$ $1.9964e-06$ $53.4408\ s$
 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 $22887$ $1.3548e-07$ $1.9964e-06$ $53.4408\ s$
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 $22887$ $1.1405e-08$ $1.2477e-07$ $95.6868\ s$
 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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