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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

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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  • 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).

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

    Citation:

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  • Figure 1.  Vertex-centred cell in 2D

    Figure 2.  The measurement stations on the Grand Casablanca area

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

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

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

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

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

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

    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.
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV
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