This is probably an ignorant question, but I've taken a couple of numerical methods courses and courses that involve using numerical methods to solve ODE/PDEs. In every course, finite difference schemes were used heavily. But it seems these numerical differencing schemes typically result in ill-conditioned discretized problems (something that I do not recall learning in any of my classes). If so, then what exactly is their use if they lead to ill-conditioned problems?
Take for example the following ODE: $$\frac{d^2u}{dx^2}=2$$
When discretized with central differencing and written in matrix form: $$Au=2$$ where A is the tridiagonal coefficient matrix resulting from the central differencing [1, -2, 1]. This is probably the "simplest" approach that I can think of to solve this problem numerically, but as the dimensions of A become bigger, this problem becomes more and more ill-conditioned.