I am solving a coupled system of non-linear PDEs in 1D. Something like,
$$ u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w) $$
where each variable is a function of $x$ (the spatial dimension).
So for each point in space I wish to calculate the Jacobian of the vector,
$$ \boldsymbol{F} = \pmatrix{F_1(u,v,w) \\ F_2(u,v,w) \\ F_3(u,v,w)} $$
Question
Should I use the PDEs themselves as a starting point to calculate the Jacobian, or is it better to use the discretised form of the equations?
I ask because the equations are very unstable without using exponential fitting, so the discretised form is quite non-trivial. My intuition is that Jacobian should follow from the discretised form as this seems more consistent. Is there a standard approach?
For example, if I were to start with the discretised form, I would simply partially differentiate each term with respect to the appropriate variable.