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.

  • $\begingroup$ First, finite difference schemes are probably the easiest schemes to understand and code. So it makes sense that you would learn these in numerical methods courses. However, you seem to be making a very broad statement when you say that 'numerical differencing schemes typically result in ill conditioned discretised problems' which I certainly don't agree with it. It would be better if you add to your post things like what kinds of problems you have found that are becoming ill conditioned, are the solutions highly oscillatory, what schemes you implemented, did you need a CFL condition etc. $\endgroup$ – Mattos Dec 7 '17 at 5:31
  • $\begingroup$ I will add something simple in the original post. I was reading some Google'd links that stated finite difference schemes are inherently ill-conditioned. $\endgroup$ – David Dec 7 '17 at 5:33
  • $\begingroup$ Post the links too if you can. It will help streamline the conversation. $\endgroup$ – Mattos Dec 7 '17 at 5:35
  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – David Dec 7 '17 at 5:37
  • $\begingroup$ dmpeli.math.mcmaster.ca/Matlab/Math4Q3/Lecture3-1/… $\endgroup$ – David Dec 7 '17 at 5:38

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