Compute advecting flux

Advecting fluxes \(P\vec{u} = \rho \theta \vec{u}\) and \(P\vec{v} = \rho \theta \vec{v}\) are computed on the cell faces.

A six-cells stencil is used with weights given in figure 1.

First the values of \(\rho \theta \vec{u}\) and \(\rho \theta \vec{v}\) are calculated in each cell. Then, since the values along the faces are calculated for all grid cells, this is equivalent to a convolution of the six-cells stencil across the entire grid with the appropriate weights kernel. This is achieved by the scipy.signal.convolve2d() function.

Note that since the convolution uses cell-centered values to calculate values along cell-faces, the output array is one index value smaller than the input array.

Fig. 5 Stencil with weights used to calculate the advecting fluxes along the cell faces.