    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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2557-2570. doi: 10.3934/dcdss.2020400
##### 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.   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.   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.   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.   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.   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.  J.-F. Louis, A parametric model of vertical eddy fluxes in the atmosphere, Boundary-Layer Meteorology, 17 (1979), 187-202.  doi: 10.1007/BF00117978.  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.  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.  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.  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.  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.  R. B. Stull, An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library, 13, Springer, Dordrecht, 1998. doi: 10.1007/978-94-009-3027-8.  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/. 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:
  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.   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.   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.   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.   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.   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.  J.-F. Louis, A parametric model of vertical eddy fluxes in the atmosphere, Boundary-Layer Meteorology, 17 (1979), 187-202.  doi: 10.1007/BF00117978.  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.  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.  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.  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.  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.  R. B. Stull, An Introduction to Boundary Layer Meteorology, Atmospheric Sciences Library, 13, Springer, Dordrecht, 1998. doi: 10.1007/978-94-009-3027-8.  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/. 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.   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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