The solution of the difference diagram for some partial differential equations from the Fourier transform and Fourier analysis can be written in ${U_{n}}^{k}=q^{k}e^{in\xi}$ form. The condition for Von Neumann stability analysis is that $|q|\leq1$. Then we can say that the difference scheme is stable. If this inequality is satisfied under a certain condition, this difference scheme is called conditional stable difference scheme. $$\frac{{U_{n}}^{k+1}-2{U_{n}}^{k}+{U_{n}}^{k-1}}{\tau^2}=\frac{1}{2}\frac{{U_{n+1}}^{k}-2{U_{n}}^{k}+{U_{n-1}}^{k}}{h^2}+\frac{1}{4}\left(\frac{{U_{n+1}}^{k+1}-2{U_{n}}^{k+1}+{U_{n-1}}^{k+1}}{h^2}+\frac{{U_{n+1}}^{k-1}-2{U_{n}}^{k-1}+{U_{n-1}}^{k-1}}{h^2}\right)+f(t_{k},x_{n})$$ $$ {U_{n}}^{0}=\phi(x_{n}),\frac{{U_{n}}^{1}-{U_{n}}^{0}}{\tau}=\gamma(x_{n})+\frac{\tau}{2}(\phi_{xx}(x_{n})+f(0,x_{n})), 0\leq n\leq M $$ $$ {U_{0}}^{k}={U_{m}}^{k}, 0\leq k\leq N $$ I tried to apply Von Neumann stability analysis for the difference scheme of the second order hyperbolic differential equation above: enter image description here.
But I don't know how to comment about $q$