Numerical solution of PDE is not really a solved problem. If your PDE involves shockwaves or turbulence (which happen for really simple PDEs) then the methods to compute solutions get incredibly fancy. There's no "black box" solution for numerical PDE, as there is for ordinary equations.
If you can solve with an analog computer, without problems with metastability, oscillations or excessive noise, then you can solve it with numeric calculus too. Those computers have much more strict limitations than digital math.
I should call the guy up sometime and get the details about what they were working. It wasn't a run of the mill analog computer, but who knows how novel it really was.
Not sure you can call any of the actual quantum computers we've made "analog", but if you're talking about the quantum annealment machines, they are generally more limited and take longer than an equivalent supercomputing cluster.
Hum... Quantum computers tend to be digital, not analog. The word "q-bit" is there for a reason, just because the bits are entangled it doesn't make them analog.
Besides, the set of problems you can solve is different from the set of problems you can solve quickly. When discussing analog and digital computers, the speed difference isn't all that relevant, but you seem to be confusing them here. Quantum computers do not solve any problem you can't solve on a classical computer, they just solve some faster.
Well yeah, that's exactly my point. Quantum computers are digital, which is not what the above comment claimed, except for quantum annealers which are closer to analog computers.
What I'm saying is that even quantum annealers aren't really faster at any practical problem than a comparable supercalculator as of yet.
The classic example are Navier-Stokes equations, a simplified model of fluid mechanics. Mathematicians do not yet know how to prove that they have a solution, but they can implement discrete versions of these equations that behave somewhat realistically. Yet, it is extremely tricky to get it right; for example, to observe turbulence at the same speed that appears on a real fluid.
A much simpler example where naive numerical methods do not work, is any first-order equation with converging characteristics. For example the inviscid burgers equation u_t + u u_x = 0. Even if you start your evolution with a smooth profile, it will form discontinuities. When you try to discretize it, computing derivatives by finite differences does not work at all, and the simplest upwind schemes have a lot of diffusion (which does not correspond to the model).
> Mathematicians do not yet know how to prove that they have a solution
Probably just a typo, but it's not hard to find particular solutions. The question is more whether solutions exist for all reasonable initial conditions. Here's a famous particular solution:
I've used a variant of this one to test CFD software.
The Millenium prize problem has a succinct statement of what's under question:
> In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
Keep in mind that this is for incompressible Navier-Stokes with a constant viscosity. For inviscid flows you could get discontinuities as you've indicated with the inviscid Burgers equations, and I know that existence and uniqueness problem for the incompressible Navier-Stokes equations has been proved for certain uncommon viscosity laws (which should be reasonably realistic as a constant viscosity is not quite right).
(And of course there are other complications not considered in this problem like boundary conditions.)
