I am trying to numerically solve the following system of partial differential equations:
$$ k^2u_{xx} - u_{tt} = f \tag 1$$ $$ c^2f_{xx} - f_{tt} = 0 \tag 2$$
I am using the method of finite differences and the central difference approximation. I know how to use the Von-Neuman stability analysis to determine the necessary stability condition in case of a single equation. However, here I have a system of equations.
I didn't find in literature how to use the Von-Neuman analysis for a system of equations. So my question is, how would I determine the stability of my system of equations using that method? Note that inserting equation $(1)$ into $(2)$ is not the way I would like to solve this problem. Also, even though I assume that the CFL condition is the necessary condition I am looking for, I would still like to derive it.