# Von Neumann analysis for coupled PDE's

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.

• If I recall correctly, you formulate the amplification matrix by appropriately expanding the error in terms of the discrete finite difference operators and then look for the conditions that yield eigenvalues within the complex unit circle. It is not always a straightforward process to find the right conditions... It depends a lot on the specific problem. I learned the process by reading other PhD thesis on the topic. I recommend reading Ch4 of Kim's thesis on fixed stress splitting methods in poroelasticity
– Paul
Sep 30 '17 at 5:15
• Thanks - I think I got caught up in discretizing everything in terms of finite difference, including the spatial parts. But the suggestion of approaching via a semi-discrete path, i.e. just use the FD explicitly for the time derivative and then represent the rest as matrix operators seems to make sense - these matrix operators have known analytic eigenvalues and so the task is 'simply' to construct the amplification matrix, which will be some linear combination of matrix products of deriv matrices and matrices representing the functions. Particularly fun if the coefficients are non-constant... Oct 1 '17 at 17:57
• What, precisely, do you mean with "a coupled problem"? Oct 26 '17 at 14:18
• I mean that I have something like $f' = a \nabla f + (b +\nabla) g$ and $g' =c \nabla g + (d + \nabla)$ $f$. Currently I had constructed the amplification matrix acting on the vector $(f,g)$, but I suppose I should consider the PDE for each individually and have the g,f as external functions, i.e. to make the amplification matrices circulant? Jul 26 '18 at 14:03