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Fluid simulations solving the hydrodynamic (HD) or the magneto-hydrodynamic (MHD) equations are very useful in physics, the latter being particularly useful for modeling plasmas.

Of course these equations are highly nonlinear and solving them accurately is difficult. My first question is, in the general case what some specific numerical challenges for building good fluid simulations solving HD/MHD?

I am also interested in the specific case of modeling multiphase gases, like those that appear in galaxy halos. What are the specific numerical challenges in this case?

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At their core, simulation of realistic fluid behavior is difficult because of the length scales involved. If you have turbulent flow, as is common in hydrodynamic or magneto-hydrodynamic situations, then you have vortices at length scales that are often vastly smaller than the size of the domain you're interested in. Resolving them both in space and time requires enormous numbers of mesh points and enormous numbers of time steps.

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Besides what Wolfgang mentions, all of your example problems have the potential for shocks which require us to capture waves that may not be present in other flow problems, and these waves can can span wide length, time, and amplitude scales which require high resolution and challenge our best numerical schemes to even converge in some cases.

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