Boundary Conditions

There are several types of boundary conditions available in artemis. Some of these are inherited from parthenon.

Choosing boundary conditions is done in the <parthenon/mesh> input block. Each boundary in the problem can be set separately.

An example input block specify periodic boundaries in the x1-direction and reflecting boundaries in the x2-direction is:

<parthenon/mesh>
....
ix1_bc = periodic
ox1_bc = periodic

...
ix2_bc = reflecting
ox2_bc = reflecting

...

Note that there are no user boundary conditions in artemis, as with some other Parthenon codes. All boundary conditions are specified directly in the <parthenon/mesh> node. Also note that some boundary conditions are only compatible with certain Problem Generators. Below is a complete listing of available boundary conditions.

periodic

periodic boundary conditions connect opposite ends of the domain.

reflecting

reflecting boundary conditions act as walls that try to keep the fluid inside the domain by mirroring variables across the boundary, and negating the sign of the normal component of vector quantities.

outflow

outflow boundary conditions attempt to let material freely leave the domain. This is not always perfect, however, and reflections can happen.

ic

ic boundaries keep the solution at the initial condition in the ghost zones. Only the disk problem generator works with ic boundaries.

viscous

viscous boundary conditions are meant to provide continuous injection at the outer radial boundary and an outflow condition at the inner radial boundary of a disk calculation. They arise from the solution of the 1D steady-state accretion disk equations which read,

\[\begin{split}\dot{M} & = {\rm const} \\\\ \partial_R ( F_\nu - \dot{M} \ell) & = 0\end{split}\]

where \(\dot{M}\) is the radial mass flux, \(\ell = R^2 \Omega\) is the specific angular momentum, and \(F_\nu = 3 \pi \nu \Sigma \ell\) is the viscous flux of angular momentum. At the innner radial boundary we enforce:

\[F_\nu = \dot{M} \ell\]

which corresponds to keeping \(\nu \Sigma = {\rm const}\) across the boundary (since \(\dot{M} = 3\pi \nu \Sigma\)). At the outer boundary, we set

\[\partial_R F_\nu = \dot{M} \partial_R \ell\]

by linearly expanding \(\ell\) and \(F_nu\). This is, therefore, and extrapolation boundary at fixed \(\dot{M}\). Once \(F_\nu\) is set, the gas density is set, and from that and \(\dot{M}\), the radial velocity is set.

The value of the injection \(\dot{M}\) is set in the <problem> block:

<problem>
...
mdot0 = 1e-6

Note that the derivation above was for 1D. 2D cylindrical simulations with planets have been shown to behave well with viscous boundary conditions. 3D simulations have been largely untested, however.

The viscous boundary condition is only allowed in the disk problem generator, and only when viscosity is active.

extrapolate

extrapolate boundaries are meant to provide an outflow boundary condition in situations where the fluid variables have non-trivial slopes. Examples include:

maintaining vertical hydrostatic balance in the strat problem generator.

maintaining the background shear profile at the \(x\) boundaries in the strat problem generator.

maintaining radial profiles of density, temperature, and velocity in the disk problem generator.

The extrapolate boundary condition is only allowed in the strat and disk problem generators.

conducting

conducting boundaries are meant to provide a constant thermal heat flux at one boundary, while fixing the temperature at the opposite boundary. This allows the domain to relax to a steady-state, hydrostatic, solution with constant heat flux. This is mainly used for internal testing. When conducting boundaries are active, the flux parameter should be set under the <problem> block. The conducting boundary condition is only allowed in the conduction problem generator.

inflow

inflow boundaries are needed for the shearing-box (See Problem Generators) in the y-direction. They are actually a combination of inflow and outflow depending on what side of \(x=0\) the boundary is on. On the inflow side, \(v_y\) is set to \(- q \Omega x\), and the other fluid variables are set to their initial values. On the outflow side, outflow conditions are set. At the lower \(y\)-boundary, inflow occurs for \(x>0\), whereas for the upper \(y\)-boundary, inflow occurs for \(x<0\). This boundary condition is only compatible with the strat problem generator.