I wish to solve an equation of the form,
$$ \frac{\partial}{\partial t} \left( \frac{\partial \phi}{\partial x} \right) = -\frac{\partial}{\partial x}(\mathcal{F}) $$
for the variable $\phi$ (e.g. mass).
On the right-hand side is the flux $\mathcal{F}$ of quantity $\phi$.
This equation "looks like" an advection-diffusion equation, but with the rate of change of the spatial derivative of $\phi$ appearing on the left-hand side.
Applying the finite volume approach we integrate the equation over the cell $\Omega$,
$$ \frac{\partial}{\partial t}\int_{\Omega}\left( \frac{\partial \phi}{\partial x} \right)dx = - \int_{\Omega}\frac{\partial\mathcal{F}}{\partial x} dx $$
$$ \frac{\partial}{\partial t}\left( \frac{\phi_{j+1/2}}{h_j} - \frac{\phi_{j-1/2}}{h_j}\right) = -\left( \frac{\mathcal{F}_{j+1/2}}{h_j} - \frac{\mathcal{F}_{j-1/2}}{h_j}\right) $$
where $h_j$ width of the cell.
Is this basic approach correct? I have never needed to solve an equation which is the time-derivative of a spatial derivative before, this is the approach I have taken, does anyone have any advice or direction? I have not yet tried to implement this numerically.