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The spatial and temporal evolution of the three species considered, as well as the maximum concentration levels, are correctly predicted by the model, although there is a certain lag between the analytical and numerical solutions (Figure 4). This lag is due to the numerical errors introduced by the discretization of the advective flux, which are reduced as the mesh size is refined. The mean absolute error (MAE) obtained with three different mesh sizes in combination with the first order and the Gamma schemes is shown in Table 1. Although the Gamma scheme reduces the MAE with respect to the first order scheme, in this case the effect of the mesh size clearly dominates the error on model output.
Table 1

First example: MAE computed from the concentration profiles 1,000 s after the beginning of the pulse for different mesh sizes (Δx) and numerical schemes

MAE (mg/L)
Δx (m)Numerical schemeorg-NNH3-NNH3-N
10 First order 0.0052 0.037 0.0145 
First order 0.0033 0.0242 0.0095 
2.5 First order 0.0025 0.0182 0.0070 
10 Gamma 0.0047 0.0349 0.0134 
Gamma 0.0031 0.0233 0.0089 
2.5 Gamma 0.0023 0.0179 0.0069 
MAE (mg/L)
Δx (m)Numerical schemeorg-NNH3-NNH3-N
10 First order 0.0052 0.037 0.0145 
First order 0.0033 0.0242 0.0095 
2.5 First order 0.0025 0.0182 0.0070 
10 Gamma 0.0047 0.0349 0.0134 
Gamma 0.0031 0.0233 0.0089 
2.5 Gamma 0.0023 0.0179 0.0069 
Figure 4

First example. Analytical and numerical concentration profiles for the three nitrogen species 1,000 s after the beginning of the discharge (left) and at the location x = 400 m (right). The numerical results were computed with longitudinal mesh sizes of 2.5 m and 10 m, and with the second order Gamma scheme.

Figure 4

First example. Analytical and numerical concentration profiles for the three nitrogen species 1,000 s after the beginning of the discharge (left) and at the location x = 400 m (right). The numerical results were computed with longitudinal mesh sizes of 2.5 m and 10 m, and with the second order Gamma scheme.

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