In order to interpolate data from the coarse grid to the fine one in the best possible way, a "conservative" interpolation formula has been developped.
We chose a local second order scheme using three contiguous coarse data points to infer data in the fine cells corresponding to the center coarse cell. The parameters of the parabolic function have been chosen in order to insure that the mean value of the interpolated data on the fine cells corresponding to a coarse cell is equal to the data in this coarse cell. Imposing this condition on three successive coarse points yields the value of parameters for the parabolic interpolation formula.
If we denote the quadratic interpolation, we have :
And the conservation of mean value conditions on three successive cells give, according to the notations of figure 4.2 :
Figure 4.2: Notations for the interpolation formula
Introducing the expression of the interpolation formula in the conservation conditions yields the value of the interpolation parameters a, b and c :
We can see that those parameters are similar to those we would have obtained had we simply interpolated the values of the coarse data on three successive points except that the independant term c differs by a second derivative-like quantity. This means the "simple" quadratic interpolation curve is smoothed somewhat with the introduction of our mean conserving conditions.
Supposing we have 2N+1 fine cells corresponding to one coarse cell, the fine cell interpolated values are given by :
We then can check that the mean value on a coarse cell is conserved :
Figure 4.3 shows the result of this interpolation formula on four different Fourrier modes on the extent of one coarse cell. Frequencies of the presented modes are lower or higher than one coarse cell's size.
Figure 4.3: Results of the interpolation ; stars indicate the interpolated fine values and + signs mark the corresponding coarse cell value.
Additionally, an interpolation normally to the boundaries must be performed in the case of scalar variables. Effectively, due to the choice of grid type, the scalar points at the boundary of the nested model don't match with scalar points of the coarse model. We have than decided to perform a linear interpolation normally to the boundary when necessary. This interpolation scheme is represented at figure 4.4.
Figure 4.4: Linear interpolation normally to the boundaries