Cannot understand why there is a factor h/2

[HannoSpreeuw]

I have probably been staring at the equations for too long, but why are there factors h/2 in the FV implementations for calcium ions, carbonate ions and the porosity?

I do not see that extra h/2 in formula 8.72 from Boudreau's Diagenetic models and their implementation, just a 2 * h in the denominator.

[HannoSpreeuw]

The same type of question, but unrelated, only about the deepest nodes:
Why do we have h**2 in the denominator instead of h for the rhs of c_ca and c_co, for the spatial derivatives of products of porosities and ion concentrations, i.e. corresponding to the second terms on the rhs of equations 42 from L'Heureux?

So my question is about h**2 instead of h on this line and this line, I probably lost track of part of the scheme's logic.

[HannoSpreeuw]

Wait, the h**2 I think I know understand.

[HannoSpreeuw]

But I am still seeking some directions in explaining how the right-hand sides for the concentrations and the porosity away from the bottom turn into the right-hand sides at the bottom.

I.e. how does this rhs turn into this rhs at the bottom

and this rhs into this rhs at the bottom?

[csummers25]

1. The factor h/2 outside came from matching the approach with the one we used for the upwind for AR/CA. We thought it was useful to rewrite the term as the centred derivative + a numerical diffusion-like term:
   $$- u \frac{(1-\sigma) T_{i+1}+2\sigma T_i -(1+\sigma)T_{i-1}}{2 h} = - u \frac{T_{i+1}-T_{i-1}}{2 h} + \frac{u \sigma h}{2} \frac{T_{i+1}-2 T_i +T_{i-1}}{h^2}$$
   but I made a mistake, the denominator should be h**2, so that when the factor h/2 is included we recover the 2h denominator, I'll correct this in the code, thanks for spotting it!

2. The differences in the formulae for the end zones came from attempting to implement the boundary conditions as described in eqn 35 L'Heureux: 'at the bottom boundary we will assume the system has no diffusive flux'. In the case of the ions we apply a boackwards stencil for the first derivative in the diffusion term:
   $$\frac{\partial}{\partial x}\left(k\frac{\partial c}{\partial x} \right) = \frac{ \left(k\frac{\partial c}{\partial x}\right)_{i} - \left( k\frac{\partial c}{\partial x}\right)_{i-1} } {h}$$
   According to the boundary conditions, the first term on the right must be zero at i=N, so the only part remaining is the second term. We did something similar for the porosity (the details of this are in the manual tex file). Since we then started using the enforced flat derivative instead I haven't thought much about whether this is still the way we want to apply these boundary conditions.