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Keplerian Ring Test
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===================
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This is a general test for hydro stability. The setup is as follows: a central mass around which particles are arranged in a ring and set on their keplerian orbits. You can find out more about setups for SPH in the [Inviscid SPH paper](https://arxiv.org/abs/1006.1524) and for GIZMO in [Hopkins 2015](https://arxiv.org/abs/1409.7395).
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There are three implementations available in SWIFT:
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+ SPH with a non-softened central potential (Setup A)
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+ Hopkins' IC's with a potential well for the ring (Setup B)
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+ A grid with particles everywhere apart from a small central region, for which the gravitational potential is softened (Setup C).
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We will dive into each of these setups below.
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Setup A: Inviscid SPH
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---------------------
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This setup is based on the one presented in the Inviscid SPH paper. Particles are distributed on a ring in concentric circles as to ensure that there is no tied-up energy in their arrangment; this situation is similar to a 'glass file'. This setup does not work with GIZMO as there is a large 'hole' in the centre (see plots below).
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Setup B: Hopkins IC's
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---------------------
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Along with the GIZMO code, Hopkins provides several initial conditions for tesitng the code, and some internal functions/ifdefs that must be turned on for each of the tests. The Keplerian Ring is ran in GIZMO with a potential well, which has the following form:
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$$
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$$
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The ring, then, is trapped within a potential well - this gets past the issues that GIZMO has with vacuous regions *but* can be seen as invalidating the test. It is noteworthy that this is *not* the setup that Hopkins describes in his 2015 paper.
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Setup C: Hopkins 2015
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---------------------
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This setup is the one that Hopkins actually describes in his 2015 paper. The ring is set up on a grid, with particles continuing all the way to the centre. This presents a bit of a problem with the timestepping for central potentials in SWIFT, and so we soften the potential at the centre, i.e. give it a form $\propto 1/(r^2 + \epsilon^2)$ with $\epsilon$ some small number. We also remove the particles near the centre, creating a small vacuum, but this (so far) has not presented any problems with the GIZMO implementation in SWIFT. |
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