We have parabolic 2D pde
\begin{align*} v_t &= \nu (v_{xx} + v_{yy}) + F(x,y,t), \; \; \; (x,y) \in R, \; t >0 \\ v(x,y,t) &= g(x,y,t), \; \; \text{on} \; \partial R, \; t>0 \\ v(x,y,0) &= f(x,y), \; \; (x,y) \in \overline{R} \\ \end{align*}
We want to study the stability of scheme
\begin{align*} \left(1 - \frac{ r_x}{2} \delta_x^2 - \frac{ r_y}{2} \delta_y^2 \right) u_{jk}^{n+1} &= u_{jk}^n + \frac{ \Delta t}{2} (F_{jk}^n + F_{jk}^{n+1} ) + \frac{ r_x }{2} \delta_x^2 u_{jk}^n + \frac{ r_y }{2} \delta_y^2 u_{jk}^n\\ u_{0k}^n &= g(0,k \Delta y, n \Delta t ) \\ u_{M_x k}^n &= g(1, k \Delta y, n \Delta t) \\ u_{j 0 }^n &= g(j \Delta x, 0, n \Delta t) \\ u_{j M_y }^n &= g(j \Delta x, 1 , n \Delta t) \\ \end{align*}
the Crank-Nicolson scheme. It is supposed to be uncondionally stable. MY question is,
Do we just need to apply discrete von neumann criteria $$ u_{jk}^n = \xi^n e^{ijp \pi \Delta x + i k q \pi \Delta y } $$
with exclusion of $F$ source term then get and equation for $\xi$
so all we need to find is conditions on $r_x, r_y$ that makes $|\xi | \leq 1$ which is the necessary condition for stability.
IS this correct?