@Kitonum My doubts were satisfied by your explanation of the faults of value​​.​​​​ I thought that it was giving wrong results for this particular PDE system. But I think that you're saying that it always gives 0 for derivatives.

@acer I don't know if this is still true in the latest Maple, but I recall a few years ago (no more than 7) that it was often necessary to do Threads:-Sleep(2) (or some other time-wasting command) after plotsetup ​​​​​​to give the GUI and kernel time to synchronize.
 

@Melvin Brown When you're through with the plot, it'd probably be best to send it to the garbage collector:

P:= 'P':

This is just to reduce the kernel's memory usage; it doesn't affect the GUI.

The Post Reply that includes my compiled tridiagonal code is here: https://www.mapleprimes.com/posts/210623-A-Finite-Difference-Scheme-For-The-Heat

@jalal How do you want to improve that animation? Do you want it to evolve more slowly? Then increase frames.

Does the above do what you want? Why haven't you responded, especially considering all the buildup?

@Kitonum Given that the value export from the solution module gives a wrong result for these (0), I'm somewhat suspicious of your results. I'm not saying that they're wrong! It would be nice if the same results or close could be obtained by a somewhat "computationally independent" method, perhaps by using smaller spacestep and timestep.

@Ajla No worries.

I showed you only what I thought was the most direct answer to your Question, but there are many possible variations of that and many much-more-sophisticated ways to numerically solve ODE systems and to present those solutions (usually as plots). So, if you're still having trouble or need any more information, let us know.

@Joe Riel I was aware of the ability to pass the list directly to Iterator:-Permute. I didn't use that because the ordering of the permutations thus produced depends on the lexicographic order of the list entries--as if they were a set rather than a list. I consider that a bug.

@Stretto I did not understand what you wanted from the grid lines from your original Question. You must admit that "grid lines [that] nicely project from the function" is nowhere near as precise as what you said immediately above. What is the purpose of the eight corner pieces on your grid lines---the parts that are essentially perpendicular to the main grid liines? Please show the code for the gridlines above, which are much better than those in your original Question.

@Joe Riel Thanks, I didn't know that an rtable could be indexed by another rtable or by NULL (rather than ..).

Is the intermediary B necessary for efficiency? Because the following seems to do the same thing:

A:= Array([a,b,c,d,e]): # Array of elements to permute
for p in Iterator:-Permute(numelems(A)) do
    printf("%{}a\n", A(p));    # print (replace with something useful)
end do:

For the tough cases, a Monte Carlo algorithm for piecewise linearity would be to evaluate the second derivative at a few randomly chosen values of the with-respect-to variable. 

@vv Your example thus makes it clear that the overall question of linearity is in general undecidable because the boolean condition in the piecewise could be made undecidable.

@Scot Gould No worries; typos happen to us all. On MaplePrimes, I can almost always figure out the intended meaning, but this time the "is" threw me off (especially since there is an is command in Maple).

@Scot Gould Thank you, I understand now. Your original second paragraph refers to an "is" feature. I'm guessing that that was a typo (which you can edit from the "More" pull-down in the bottom right corner).

@Scot Gould I agree. Explore is a great feature. And it can also be used for commands other than plots. 

Would you please edit your second paragraph for clarity?