I have a set of rather nasty, nonlinear difference equations roughly of the following form:
$ \frac{a^{(j)}_{i+1}(s)-a^{(j)}_i(s)}{\epsilon}=f^{(j)}(\lbrace a^{(j)}_i(s)\rbrace_{j=1}^m,\lbrace a^{(j)}\prime_i(s)\rbrace_{j=1}^m,\epsilon) $
I have one such equation for each $j$ from 1 to $m$. Here $a^{(j)}_i$ are functions of $s$ for each $i,j$, and primes denote derivatives with respect to $s$. The subscript $i$ is like a discrete time variable.
[I actually have a few problems that I can put in this form so I am curious what can be said at this level of generality.]
This set of equations propagate the initial data $\{a^{(j)}_0(s)\}_{j=1}^m$ so long as the $f^{(j)}$ remain well-defined (e.g. no dividing by zero or anything like that), and I can write down some set of inequalities on $\{a^{(j)}_i\}_{j=1}^m,\{a^{(j)}\prime_i\}_{j=1}^m,\epsilon$ so that this is true at any given step.
In the limit $\epsilon\rightarrow0$, I can expand the right hand sides in a Taylor series in powers of $\epsilon$ and I can formally write:
$ \frac{\partial a^{(j)}(x,s)}{\partial x}=f^{(j)}_0(\lbrace a^{(j)}(x,s)\rbrace_{j=1}^m,\lbrace\partial_sa^{(j)}(x,s)\rbrace_{j=1}^m)$
Where $f^{(j)}_0$ is the first terms of the Taylor series for $f^{(j)}$, respectively, and $a^{(j)}(x,s)$ is supposed to be like a continuum limit of $a^{(j)}_i(s)$ as the $i$ spacing gets small. The initial data for this is now a set of functions $\{a^{(j)}(0,s)\}_{j=1}^m$.
I am not an analyst at all so maybe this is either trivial or impossible, but
1) I would like to know whether or under what circumstances solutions of the difference equations converge to solutions of the differential equations, and how to show this convergence.
2) It turns out that when I take $\epsilon\rightarrow0$, the inequalities that guarantee solution to the difference equations are always satisfied - does this imply that the differential equations won't generate singularities as I evolve in $x$ as well?
I looked a bit in the literature but I mainly found articles going the opposite way, i.e. looking for good discretizations of PDEs, rather than showing that particular discretizations converge. Maybe I am missing something?
Feel free to recommend some books or articles or even "magic words" for google if this is all really basic.