Suppose I have a PDE $$ \frac\partial{\partial t}f(x,t) = F\left(f(x,t), \frac\partial{\partial x} f(x,t)\right). $$ I would like to discretize $f(x,t)$ as $$f^\epsilon(x,t) = \sum_{n=0}^{N_\epsilon} f^\epsilon_n(t) \chi_{[n\epsilon, (n+1)\epsilon)}(x), $$ where $t$ is kept continuous, and replace partial derivatives with respect to $x$ by finite differences. The resulting system of ODE's then looks like this: $$ \frac{\mathrm d}{\mathrm d t}f^\epsilon_n(t) = F^\epsilon\left(\{f^\epsilon_n(t)\}, \{\Delta^\epsilon_n f^\epsilon_x(t)\}\right) $$ with $F^\epsilon\rightarrow F$ and where $\Delta^\epsilon_n$ is, e.g., a forward difference operator. I want to solve this system of ODE's for all $f^\epsilon_x(t)$ and then take the limit $$g(x,t)=\lim_{\epsilon\rightarrow 0}f^\epsilon(x,t).$$
What conditions can I impose on $F$ in order to show that $g$ solves the original PDE? Does the theory of finite difference methods in PDE's help me in any way? After all, it's much like a finite difference scheme, only with the time variable left continuous.