3 ms·

Can you provide a simple example of this?

enriquto 6y ago
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).
- btrettel 6y ago
  > 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: https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex 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. https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc... 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.)
  - enriquto 6y ago
    > Probably just a typo, but it's not hard to find particular solutions. Indeed, a fluid eternally at rest is surely a solution :)

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).

btrettel 6y ago
> 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: https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex 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. https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existenc... 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.)
- enriquto 6y ago
  > Probably just a typo, but it's not hard to find particular solutions. Indeed, a fluid eternally at rest is surely a solution :)