I know if I have say $v_t + a v_x = 0$ I can write it as $Lv = 0$ where
$$ L = \frac{\partial}{\partial t} + a\frac{ \partial }{\partial x} \; \; \; \; \; and \; \; \; \; v=v(x,t)$$
Now, Given a difference scheme, say
$$ u_k^{n+1} = u_{k}^{n-1} - \frac{a \Delta t}{\Delta x}( u_{k+1}^n - u_{k-1}^n ) $$
In my books, I am given that a scheme is consistent if $L_k^n v_k^n - G_k^n \to 0 $ and $\Delta t, \Delta x \to 0$.
Im trying to understand the notation of the sequence. If I rewritte the scheme I have
$$ \frac{u_k^{n+1} - u_k^{n-1} }{\Delta t } + a \frac{ u_{k+1}^n - u_{k-1}^n }{\Delta x} = 0 $$
Which looks as our pde. I am having hard time trying to visualize what is $L_k^n$, $v_k^n$ and $G_k^n$ in this scheme. Can someone point me in the right direction? best regards