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Topic: Re: PDE discretization; what happens with PDEs?
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Alexei Boulbitch

Posts: 483
Registered: 2/28/08
Re: PDE discretization; what happens with PDEs?
Posted: Dec 6, 2012 4:55 AM
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Is there any way to figure out what NDSolve default spatial discretization
is for a given set of initial conditions? The notes state that it is "based on the initial conditions", but I cannot figure out a set of options that will let me see spatial locations at which the PDE is actually being solved.

Motivation: The default discretization is yielding results that I cannot replicate using "Coordinates" to fix my discretization.

Any assistance is appreciated.

First to address your question: try to have a look at Menu/Help/Advanced Numerical Differential Equation Solving in Mathematica/Partial Differential Equations/Controlling the Spatial Grid Selection
You may fund some useful hints there.

I would like, however, to ask a further question.

At the Mathematica conference in London this Summer Mr. Stephen Wolfram informed us that PDE solving by NDSolve will be included in Mathematica 9. Mr. Wolfram highlighted this point in his talk, when he explained how much M9 differs from M8. I understood this message the way that some important advancements are achieved in this area.

Just to remind, Mathematica 8 was already able to solve certain types of PDEs. It used the so-called, Method of Lines. The latter may only be applied, provided the PDE may be formulated as initial+boundary problem. A natural example of such a formulation is the time-dependent diffusion equation. M8 was unable to solve pure boundary problems, such as, for example, the Laplace equation.

My question is: if something has changed in M9 with the PDE solution and what the changes are? Does it apply something other method(s) in addition to the Method of Lines?

I checked the help. On one hand I find no indication that only the Method of Lines may be applied. On the other hand, all PDE examples in the Help/NDSolve are those I have already seen in M8.

Besides, one can directly check that Laplace equation is still not understood by NDSolve:

eq = D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 1;
bc1 = u[0, y] = u[1, y] == 0;
bc2 = u[x, 0] = u[x, 1] == 0;

NDSolve[{eq, bc1, bc2}, u[x, y], {x, 0, 1}, {y, 0, 1}]

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>

The error message here look pretty much like the indication that only the previous paradigm of the initial+boundary value problem can be understood by the NDSolve.

Finally, so far I have only found those tutorials on PDEs that have been there since M8.
Have I missed something?

I would be grateful for any explanation of what is the current status with PDE equations.


Alexei BOULBITCH, Dr., habil.
ZAE Weiergewan,
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