Symmetry and physical boundary conditions

Symmetry boundary conditions are those that arise by viewing the computational domain as a subregion of some larger domain which possesses symmetries. These symmetries allow a simulation of the subregion to act as an effective simulation of the larger encompassing domain, because the latter can be inferred from the former via the symmetry. For example, one can often simulate a rotating star by `slicing' the space in half through the equatorial plane, simulating only one half, and placing a reflection boundary condition at this plane. The symmetry can be regarded as a property of the underlying computational grid upon which the simulation takes place.

It is often possible to describe the symmetry of a physical problem in terms of multiple `simpler' symmetries. Going back to the rotating star example, we can note that there is also a rotational symmetry about the axis of the star. Thus it is sufficient to simulate only the upper half of a $\phi=\mbox{const}$ plane of the star, since rotational symmetry will recover half of the star from the single plane, and the reflection symmetry can recover the other half of the star. To do this we use two symmetry boundary conditions, one for the rotational symmetry and one for the reflection symmetry. At the edges and corner grid points there will be two symmetry boundary conditions active, which illustrates a general point about symmetry boundary conditions, namely that there can be any number of them active at any given grid point. In addition symmetry boundary conditions are often non-local, for example a periodic boundary condition which applies in simulating plasma in a tokamak.

Physical boundary conditions are motivated by the physics of the quantity that the grid variable represents, such as one which allows outgoing waves of a scalar field to propagate off the grid, but does not allow ingoing waves or reflections. Usually the same physical boundary condition is applied to all external boundaries of the computational domain, however this is not always the case. Currently thorn Boundary allows a separate boundary condition to be applied to each face of the domain, however this is only implemented at the moment using the older deprecated interface. Face specific calls will be available using the current interface shortly. It is also possible that one will want to use different physical boundary conditions at different regions of a face, and support for this can be added if necessary. Usually physical boundary conditions are local. A non-local physical boundary condition may arise e.g. from a need to solve an elliptic equation at the boundary. As opposed to symmetry boundary conditions, it only makes sense to have a single physical boundary condition active at a given grid point.^part11

To summarize, a `physics' thorn, such as a spacetime evolution thorn, knows only about physical boundary conditions. Symmetry boundary conditions are aspects of the grid and are managed by other thorns.