In FVM, we have to compute fluxes at some face of a cell. There are many ways to compute this face flux value, but the most common and easiest way involves some simple averaging at the face. However, I am a little confused on what the best way of averaging is.
The particular case I am wondering about is when you have some material property multiplied by a gradient. For example, the heat conduction equation is $k \frac{dT}{dx}$.
For this, we want to compute $k \frac{dT}{dx}$ at a face. By using the simple averaging technique, I can see two ways you can do this.
To illustrate the two ways, let's assume that the cell $L$ and cell $R$ are the neighbors of the face. Let's assume the cell centers are of equal distance from the face centroid, to avoid talking about weighted averages.
The two ways are:
(1) $$ 0.5(k_L\frac{dT}{dx}|_L + k_R\frac{dT}{dx}|_R) $$
(2) $$ 0.5(k_L + k_R) * 0.5(\frac{dT}{dx}|_L + \frac{dT}{dx}|_R) $$
In the first case, we compute the heat conduction in each cell and then average it to get the heat conduction at the face. In the second case, we average the conductivity in each cell, and also the temperature gradient in each cell and then we compute the heat conduction at the face from the averaged conductivity and average temperature gradient.
Is one method superior vs. the other? Obviously, both methods are equivalent if $k_L = k_R$, so this discussion assumes they are different.