I am considering two equations $$u_t=a(x)u_{xx}$$ and $$v_t=b(x)v_x$$ as classical representatives of the parabolic and hyperbolic family of equations. If $a(x)=a$ and $b(x)=b$ were constants, to show stability of any finite difference scheme I use for those equations, I could use the Fourier or so called Von Neumann stability analysis, i.e. would have to calculate the magnitude of the amplification factor.
In the case of variable coefficients though, where $a=a(x)$ and $b=b(x)$, I am somewhat confused. From the book by Strikwerda (Finite Difference Schemes and PDEs) on page 59 it says that one can use method of frozen coefficients, i.e. freeze the values at every grid point, use the Fourier analysis and deduce stability for variable coefficients. On the other hand, in the book of Ascher (Numerical methods for Evolutionary Differential Equations), on page 153 it says that even ff the method is stable with constant coefficients then it doesn't necessary imply stability for variable coefficients.
Thus I see a contradiction between two books. My problem is that I see how the von Neumann analysis works and I wonder if I can use that with variable-coefficient PDEs as well. If I can, is that the same for both of the equations I am looking at or there are some differences between the two in terms of the method of frozen coefficients? Basically, where is the limitation of the Von Neumann stability analysis for parabolic and hyperbolic equations?