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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.

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If you need Jacobian matrix information for a numerical method, you should calculate the Jacobian matrix of the discretized form of the equations, since that will be consistent with the discretized equations you are solving.

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    $\begingroup$ As a comment, you can see this as a larger issue for schemes with stabilization (shock capturing terms in CFD as an example). I'm not sure if your specific monotone/exponentially fitted scheme falls under this umbrella, but if it does, it's often advantageous to include these additional terms in the Jacobian as well due to their nonlinear effects. Jacobian-free Newton-Krylov methods (approximating a matrix-vector product by a difference scheme) often cite this as an advantage, that they automatically account for all such nonlinearities in a discrete scheme. $\endgroup$ – Jesse Chan Aug 12 '13 at 17:37

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