Accuracy estimation.

We get a good estimate of the error if we first assume a pointwise approximation to the exact solution: $S_g = L + A (\Delta x_g)^p$. The pointwise error estimate is then: $E_g= S_g - L = A \; (\Delta x_g)^p$, where A is a constant value, g denotes the grid being used, S_g is the obtained solution, L is the exact solution and p is the unknown error order.

The considered grid sizes follow the relationships: $\Delta x_{th5}=4 \Delta x_{th80}$ and $\Delta x_{th20}=2 \Delta x_{th80}$, then, using grid th5, th20 and th80 in the refinement study:

$$\begin{split} & S_{th5} = L + 4^p A (\Delta x_{th80})^p\\ ... ...80})^p\\ & S_{th80} = L + A (\Delta x_{th80})^p \end{split}$$ (10.1)

Being S_th80 the numerical solution with the smallest truncation error, we can approximate the exact solution L by it. By evaluating the ratio between two successive error estimates E_th5 and E_th20 we can estimate p, the unknown order of the approximation error.

From the above three equations, eliminating L, A, $(\Delta x)^p$, and taking the norm, we get the ratio R_a=|| S_th5-S_th80 || / || S_th20 - S_th80 || = (4^p-1)/(2^p-1). Solving for p gives: p = log₂( R_a-1).

The solution tested included the level $\eta$. Accuracy estimation in the velocities u, v is the same.

Figure 7.1 represents level output at 3 points of the channel: P₁ at the West border (inflow, x=0), P₂ at the middle of the channel (x=100 km) and P₃ at the East (outflow, x=200 km) border. At this ordinate axis scale output from the model running on th5, th20 or th80 shows no appreciable difference. It may be argued that the most significant point to measure the accuracy is P₃ where the non reflective BC is being applied; however, we decided to use all 3 points along the channel to be more general.

Figure 7.2 shows the difference of the level output running over the th5 ($\Delta x$=2000 m) and th80 ($\Delta x$=500 m) grids. Figure 7.3 shows the difference of the level output running over the th20 ($\Delta x$=1000 m) and th80grids. Notice the different y axis scale when compared to figure 7.1.

Table 7.1 represents the numerical values obtained in the above mentioned simulations. The infinite norm is employed.

 

Table 7.1: Numerical accuracy order obtained by the grid refinement study.

Point	$\Vert S_{th5} - S_{th80} \Vert$	$\Vert S_{th20} - S_{th80} \Vert$	$\frac{\Vert S_{th5} - S_{th80} \Vert} {\Vert S_{th20} - S_{th80}\Vert}$
P1	$0.842\times10^{-3}$	$0.101\times10^{-3}$	8.33
P2	$0.912\times10^{-3}$	$0.138\times10^{-3}$	6.60
P3	$1.006\times10^{-3}$	$0.136\times10^{-3}$	7.40

 

The analytical accuracy presented in Section 7.1 suggests that the numerical scheme is at least second order accurate while this experiment shows a slightly better behavior instead.

  

Figure 7.2: Pointwise error E_th5.

  

Figure 7.3: Pointwise error E_th20.