Why are there researchers in PDE theory when we can instead build numerical solvers?

Question

I have a bit of a naive question:

Why are there researchers in PDE theory, e.g. people who work on analysis and PDEs, when instead one can spend their research days building numerical solvers to get solutions that are as close to an exact answer as we want?

I could see how maybe 50 - 100 years ago, without computers / computing power, PDE researchers would be theorists and pure mathematicians, but why are there still pure PDE researchers today?

I know many of them win big awards / get lots of recognition, but I don't really understand what they do and why it's important, when we have so much computing power today.

Answer 1

It's saddening to read such a question from an "aspiring $\ldots$ scientist" (according to his profile).

Why should we do trigonometry, or some other kind of geometry, if we can measure lengths or angles with any desired precision using a yardstick or a protractor?

Aren't you aware that the "praised PDE researchers" dig up universal truths, valid for all PDEs of this or that kind, whereas numerical solutions in most cases are about a single problem instance. E.g., how could you find out (let alone prove) by numerically solving millions of ODE's to the highest precision that under certain geometric circumstances there has to be a periodic solution?

Answer 2

I encourage you to take up the task of writing a numerical PDE solver. This will open your eyes to the answer to this question.

First thing that will happen: You won't know if your solution is correct. So you'll scour the literature for analytic solutions, and look for a PDE theorist to help you there.

Next, your solver will be unstable. Well, certainly it isn't guaranteed that this is a problem a PDE theorist can help you with (a numerical analyst might be your friend there), but perhaps Peter Lax could tell you about the formation of shockwaves and help you detect when a numerical solution is going to be useless.

Then, you'll notice that your solution really isn't all that good, even though it isn't useless. You'll have to ask a PDE theorist about conserved quantities, and try to develop numerical methods that respect these conservation laws, such as is done in symplectic integrators for ODE solvers.

The inside of a CPU is a dark place, and any flashlight you can use to illuminate whether it's doing the right thing is invaluable. Thank God for the PDE theorists!