I am interested in understanding how to perform stability analysis for coupled (to keep things do-able, lets say linear) PDE's
In the case of a single PDE, i understand the logic behind the VN analysis: mode expansion and look at the amplification factor of any given mode.
However for coupled PDE's it seems like you won't be able to cancel things as nicely.
For example, basing off the notes here https://en.wikipedia.org/wiki/Von_Neumann_stability_analysis
specifically starting at Eqn (2): If this represents the error for one function it will depend on the other via the coupling. It seems that the $e^{at}$ term will not cancel nicely to get Eqn (6). Presumably i can still restrict the analysis to one mode at a time assuming both PDEs are linear?
That is, rather than the $e^{at}$ terms canceling as they do in that simple example, in the coupled case i would be left terms containing the relative size i.e. $e^{(a-b)t}$ where $a$ and $b$ are the exponents of the mode expansion amplitudes of the errors for the two PDE functions respectively (whoa, a mouthful...)
It seems like the solution is to cast as a linear matrix problem but I get a bit stuck here too. Certainly one can write the time discretization in terms of an amplification factor/matrix $A$ acting $n$ times on some initial vector to give the new state, but how does one then look at the spatial discretization (contained inside the amp matrix $A$ in this context? Or do we just look for eigenvalues of that amplification matrix and then do finite-difference afterwards?
Any and all help is appreciated.