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I have always been fascinated by the so called taxicab geometry first considered by Hermann Minkowski. The metric that has to be used here is a L1 distance which e.g. means that the lenght of the diagonal in a unit square is 2 and not √2 - and this holds true no matter how fine the mesh is! Basically the solution doesn't converge when approximated dicretely.

If you see the mesh as an analogy of solving a differential equation numerically the difference between their true solution and the numerical outcome is obviously huge and won't be acceptable in most applications.

My questions: Do you know of certain classes of differential equations where no closed form solutions exist AND that misbehave in an corresponding way? How are they called and where do I find more information on those? How do you identify them (what characteristics do they have) and how do you tackle such misbehaved equations?

Addition: When I think about it: What could be even stranger is the case when a closed form solution exists but the differential equation can't be solved numerically in the abovementioned sense.

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  • $\begingroup$ I don't understand your first paragraph - as you say, the length of the diagonal curve of a square is 2 in the taxicab metric. So why would you say that "the solution doesn't converge". The sum of distances of subdivisions of the diagonal certainly converges to 2 in the taxicab metric. $\endgroup$
    – j.c.
    Commented Oct 29, 2009 at 18:33
  • $\begingroup$ increasingly finer subdivisions, I mean $\endgroup$
    – j.c.
    Commented Oct 29, 2009 at 18:33
  • $\begingroup$ What I mean is that when you try to evaluate a length in a continous setting by a discrete approximation. It simply stays at 2 and doesn't move towards Sqrt(2). $\endgroup$
    – vonjd
    Commented Oct 29, 2009 at 18:45
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    $\begingroup$ All that means is that length isn't a continuous function in whatever topology you're imposing on curves. I don't see what this has to do with differential equations. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 29, 2009 at 18:54
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    $\begingroup$ Probably the granular media are best physical example of systems which do not allows easily description by differential continuous models is disputable: en.wikipedia.org/wiki/Contact_dynamics In multi body regime of contact dynamics, where You consider for example sand, there is serious problem to obtain phenomenological equations from microscopic one. Microscopic system is too overcomplicated and we believe there should be simpler macroscopic PDE. But such systems dynamic is very complicated and we have troubles to model it on the macroscopic level. $\endgroup$
    – kakaz
    Commented Feb 26, 2010 at 14:28

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You'd probably be interested in reading about discrete differential geometry, as put forth by Bobenko et al. Here's one paper about when certain geometric quantities defined in the discrete sense converge to the analogous quantities in the continuous sense. Again, I don't know what connection you're trying to make to differential equations, (perhaps the results on convergence of finite element methods?) so perhaps you should spend some time trying to make that precise and then edit your question again.

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I'm not a numerical analyst, and I didn't really understand your question, but one example that came to mind as I was reading it was oscillatory phenomena at discontinuities. If you model a linear initial value problem with a jump discontinuity using a linear method with greater than first order accuracy, you will get oscillations near the jump, and this can be qualitatively at odds with predicted behavior. It is a numerical analogue of the Gibbs phenomenon. There are ways of damping this using nonlinear methods like flux limiters.

As far as I can tell, you are asking for a fairly generic phenomenon. If you choose a random PDE with sufficient complexity, you should expect to encounter bad stability behavior and numerical intractability. For well-behaved PDEs, there are ways of rigorously bounding error.

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