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I am not really experienced in the field of numerical differential equations. I am only studying existence and uniqueness results about PDEs at the moment. Now I was wondering what would happen, if one tries to solve a PDE numerically, although the PDE 1) has no solution or 2) the solution is not unique?

The Navier-Stokes-Equation in 3 dimensions comes to mind. How do physicists approach this equation? Do they work with numerical solutions although there is no proof, that there is a solution at all?

I really appreciate your help!

Best,

Luke

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What happens depends strongly on the software to compute the solution, but in general if you don't have a proof of existence and uniqueness as well as a proof of convergence to the unique solution, you are on very shaky ground.

Let's consider $y' = f$ with $y(0) = 0$. This can be solved by an ODE solver, under some assumptions on $f$ (say, continuity). Now let's say $f$ is a delta function at a random location in the domain. A numerical solver will never find it, and the answer will always be wrong.

The next situation is non-uniqueness. I wrote a few solvers for a PDE which had many solutions. What we attempted was to compute them all, and then select a subset of the solutions based on some property we thought was relevant. (To be fair, what actually happened is we got into fights about what was relevant, and eventually we just modeled the phenomenon with a different PDE that did have a unique solution. I can honestly say that non-uniqueness was one of the biggest headaches I've had solving PDEs.)

I've never attempted to solve an equation which known to not have a solution, but I cannot imagine it turning out well. The best you can hope for is an error message, but you probably won't get it. My suspicion is that your numerical method will compute the solution to an equation which is in some sense "close" to the original, and then you'll be in a truly devastating situation where you have a bunch of garbage which you think is a solution.

Hopefully someone else can offer insight into Navier-Stokes; I've never worked on it so I can't help you there.

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  • $\begingroup$ Thank you for your comment! I read on Wikipedia, that solutions to the Navier-Stokes-Equations are used in physics, although it has not been proven that they exist. If several different solution calculators come to a (in some sense) close solution, is that a hint, that the solution might exist and be unique? Is that enough for applications? $\endgroup$ – Luke May 31 '17 at 6:48
  • $\begingroup$ Sure it's a hint. That's probably part of why people generally believe that solutions to Navier-Stokes do exist. $\endgroup$ – user14717 May 31 '17 at 14:29
  • $\begingroup$ Does that mean when I use a PDE solver such as gPROMS or COMSOL to solve multiphysics problem, just because the numerical solver converges, the results may not be unique! Doesn't the fact that you add boundary conditions and physical constraints increase the likelihood that the results are unique? $\endgroup$ – SPIL Oct 31 '17 at 11:12
  • $\begingroup$ I'm not sure "likelihood" is the correct way of understanding uniqueness. As PW Anderson says, computation should come after understanding, not before. As to the behavior of COMSOL and gPROMS, I cannot say what their behavior will be. $\endgroup$ – user14717 Oct 31 '17 at 18:18

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