For example I have an equation that goes something like
$ \partial_t \rho = -\nabla\cdot (\rho u) + \nabla \cdot(D(\rho, T) \nabla \rho) + \rho_s $
($\rho, \rho_s, u, T$ are coupled with a few other PDEs)
Could I use Forward Euler to approximate the future $\rho$ and $T$ to estimate the future coefficients and then plug them into an implicit scheme or is there a better way?
As equation of state I have table values for the coefficients $D(\rho, T)$ and am able to interpolate between the values, so I should be able to compute Jacobians of the interpolants. Can I use the Jacobian at the current time if it is not good enough with Forward Euler on PDE, so everything is implicit except for the coefficient which uses an explicit method with the Jacobian?
If that isn't good enough either I would really like some input how to proceed. How would one get the Jacobian from the future? Would I have to perform some kind of sampling for possible future values of $\rho$ and $T$?
(I want to solve the equations in scholarpedia but this toy example should be enough to get me started)