> I have tried MATHEMATICA, but my general impression is that > the error control is entirely unreliable. Even if one requests > AccuracyGoal->12 and PrecisionGoal->12, the program returns > at most 3-4 accurate digits. Requesting higher precision produces > failures due to the lack of memory, etc. > The above is probably easy to predict, since MATHEMATICA seems > to use rather standard non-adaptive finite-differences, and these usually > cannot be more accurate. So, it seems that the arbitrary precision > arithmetic is not really of advantage in MATHEMATICA, at least > in the case of the PDE solving. > > (or maybe I am wrong and someone will tell me how to overcome these > barriers).
According to your description, It sounds like that problem is caused by the method employed to get the solution, not a lack of accuracy associated with the data types used to express the solution. If a method that only produces approximate solutions is used then we can only expect to get approximate solutions from it, no matter what numerical data types are used to express the values.