# Stability of nonlinear partial differential equation

I want to find an expression for the stability of the nonlinear Poisson equation. I know about von Neumann stability analysis which applies to linear equations as far as I know. Any suggestion how to analyse the stability?

The equation I am trying to solve is:

$$-\frac{\partial ^{2}\phi}{\partial \tau^{2}}+\alpha\frac{1}{r'^{2}}\frac{\partial}{\partial r'}\left(r'^{2}\frac{\partial\phi}{\partial r'}\right) = \left(\frac{\gamma}{\phi^{n+1}}+\beta\right)\,.$$

And my discretisation looks like:

$$a\phi_{i-2,j}-2a\phi_{i-1,j}+b\phi_{i,j+1}+ c\phi_{i,j-1}+d= e \phi_{i,j}+\frac{h}{\phi_{i,j}^{n+1}}$$

$a, b, c, d, e, h$ are constants, and $i$ and $j$ indices stand for time and radius.

• If you are trying to analyze the stability of your/some numerical method to solve some equation, one needs to know about the method. Von Neumann stability analysis is for analysis of finite-difference schemes applied to PDEs. Please, reformulate your question to what exactly you want to analyze, provide all the information and what you have already tried. – Anton Menshov Apr 23 '17 at 14:18
• What does $\phi^{n+1}$ mean I'm your continuous equation? As it is written, it seems linear. But that depends on what that variable means – nicoguaro Apr 24 '17 at 12:49
• @nicoguaro -- the $n+1$st power of $\phi$ maybe? :-) – Wolfgang Bangerth Apr 25 '17 at 2:06