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So I was reading this paper: https://www.nature.com/articles/s41534-020-00291-0.

I am a bit confused about this:

Little attention, however, has been given to quantum simulation of a classical nonlinear continuum system such as a viscous fluid even though this too is hard for classical computers. Such fluids obey the Navier–Stokes nonlinear partial differential equations, whose solution is essential ...

Why are the Navier-Stokes equations difficult to simulate? That's what the above quote is saying, right? It is just because there are a lot of variables?

Thanks

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  • $\begingroup$ There are many reasons why. There are a lot of relevant variables to solve for, all of which are mixed in a system of nonlinear PDEs. Then of course there's turbulence and chaos. $\endgroup$
    – K.defaoite
    Dec 15, 2022 at 0:24

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In some cases (that is, for very viscous fluids) the Navier-Stokes equations are relatively easy to simulate. In most real-life cases, the reason they are hard is because small but non-zero viscosity leads to chaotic behaviour on spatial scales smaller than the domain size, i.e. turbulence.

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  • $\begingroup$ In most real-life cases, the reason they are hard is because small but non-zero viscosity leads to chaotic behaviour on spatial scales smaller than the domain size, i.e. turbulence.. What is "chaotic behavior"? Does this mean that the graphs go up and down a lot? If so, why would it be hard to simulate that? Thanks. $\endgroup$ Dec 15, 2022 at 2:01
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    $\begingroup$ Chaotic means not just 'up and down a lot', but highly sensitive on the initial condition (and so when simulating, is likely to change depending on numerical parameters such as spatial/temporal resolution). Imagine you are trying to model the ocean (hundreds of km) where the solution depends crucially on what is happening on the scale of cm. And if you don't resolve that scale with a lot of accuracy your answer will be wrong. Think of the resolution you'd need. That's Navier-Stokes. $\endgroup$
    – messenger
    Dec 15, 2022 at 2:15
  • $\begingroup$ Re: ocean. In fully 3 dimensions Navier-Stokes are difficult but many aspects of the ocean suggest reduced dimensionality such as the shallow-water approximation, stratification at the thermocline, and topological containment along the equator. So there may be hope to simulate the ocean. $\endgroup$
    – wehute
    Dec 19, 2022 at 22:32

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