next up previous
Next: Example Problems Up: Mathematical Method Previous: Biasing


Photon Teleportation

The equations governing the atomic populations are discretized in space and time, and one can expect some systematic errors to arise from this in a scheme where atomic populations are constant within a zone. Photon teleportation is an error that arises when the absorption mean free path is small compared to the size of a zone, and the time step size is also small (in the case for IMC). As photons stream through a zone they are attenuated by absorption. If the opacity is high, photons entering one side of a zone are completely absorbed very near the boundary. The scoring of energy deposition, however, is done as if the absorption had taken place evenly across the entire zone. On the next time step, the corresponding spontaneous emission is handled assuming this uniformity in the zone. In effect, spontaneous emission occurs too early on the far side of the zone and energy is transported across the problem too quickly. This error is aggravated in IMC if the time step size is reduced without a coordinated adjustment to the zone size.

SIMC is much more susceptible to this problem than IMC. This is due to the reduced net absorption for IMC, where a portion of the physical absorption has been converted to effective scattering. This beneficial influence of effective scattering comes at the cost of increased execution time and evaporates as the time step size is reduced. In the limit that $ \Delta$t goes to 0, IMC has the same teleportation error as SIMC. Teleportation error is properly addressed by decreasing the optical depth per zone. Unfortunately, reduction of zone size in order to reduce teleportation error leads to an increased running time as particles cross many zones in order to escape the problem.

Photon teleportation has been observed in many previous studies. In fact, it is known that a finite difference discretization that assumes constant conditions within each zone and time step does not tend to the diffusion limit of the transport equation [6,9,10,11,12,13,14]. A heuristic treatment informally known as source tilting [15], frequently used in IMC, could be used in SIMC to reduce teleportation effects. A self consistent, piecewise linear treatment of atomic population would obtain the correct diffusion limit.


next up previous
Next: Example Problems Up: Mathematical Method Previous: Biasing