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This is a very simple one-dimensional heat-conduct equation, the only special part of it is the piecewise initial condition:

    b = NDSolve[{D[tes[t, x], t] == D[tes[t, x], x, x] + Exp[-1/(tes[t, x])], 
                 tes[t, 0] == 1, tes[t, 1] == 1, 
                 tes[0, x] == Piecewise[{{-100 (x - 0.1)^2 + 2, 0 <= x <= 0.1}, 
                                        {2, 0.1 <= x <= 0.9}, 
                                        {-100 (x - 0.9)^2 + 2, 0.9 <= x <= 1}}]}, 
                {tes[t, x]}, {t, 0, 100}, {x, 0, 1}]

If you run the code, you will get this warning message:

NDSolve::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x

Why does this message come out? I read the help but I don't think my initial condition has that kind of fault: it's piecewise but smooth, right?

I'd like to add another sample here since its behavior makes a interesting contrast to the sample above:

c = NDSolve[{D[tes[t, x], t] == D[tes[t, x], x, x] + Exp[-1/(tes[t, x])],
             tes[t, 0] == 1, tes[t, 1] == 1, 
             tes[0, x] == Piecewise[{{-Exp[-1000 x] + 2, 0 <= x <= 0.1},
                                     {2, 0.1 <= x <= 0.9}, 
                                     {-Exp[-1000 (1 - x)] + 2, 0.9 <= x <= 1}}]},
            {tes[t, x]}, {t, 0, 100}, {x, 0, 1}]

This example choose Exp as the transition between initial condition and boundary conditions, in fact this initial condition is not even continous at x=0.1 and x=0.9 in the view of math, and I think the change of Exp is more drastic than the polynomial function, but it gets no warning message. Well, I should say, after all this time (notice the time I posted this question) I've already treat this as some kind of bug, but I still expect an in-depth explain for it.

share|improve this question

I still get a warning but I think it's less worrying if I use an alternative form of your initial condition :

   tes[0,x]==1+UnitStep[x-0] UnitStep[1-x]},{tes[t,x]},


share|improve this answer
Haha, yeah, in fact this is the original problem I'm trying to solve, but at last I found that mathematica seems to have a high require for…er…consistency (it seems that the concept "one-sided limit" " One-Sided Derivative" etc. are not available in NDSolve), and this is just the reason why I turned to the problem now you see. – xzczd Jul 30 '12 at 14:44

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