In this paper, we have compared Implicit Monte Carlo (IMC) to the Symbolic Implicit Monte Carlo (SIMC) technique, using collisional pumping, the surface heating problem and the Milne problem as diagnostic applications. SIMC was also extended with a temporally persistent version of the weight vector approach of [5], demonstrating the value of weight vectors when accurate spectral information is desired. In addition to the numerical runs, partial analytical solutions were used to verify the accuracy of the Monte Carlo solutions and to point to the sources of systematic errors when they occurred.
An important result of this investigation is to demonstrate conditions where SIMC and IMC succeed and fail. In general, SIMC produces much lower noise for high opacity problems, with the spectral results using the weight vector approach being truly stellar, because the SIMC algorithm does not expend computer time performing non-physical effective scattering. This performance advantage comes at the price of increased sensitivity to teleportation error that results from high opacity per zone and from additional time needed to solve a linear system of equations with matrix size based on the number of zones. Although teleportation error will also occur for IMC, the level of severity is moderated by the portion of the physical absorption that is rolled over into effective scattering. Finer zones are required for SIMC than are needed for IMC as a result.
While the effective scattering in IMC dampens the effect of teleportation error, it becomes highly subject to this problem when the time step is small. When effective scattering is small, sensitivity to teleportation error rises to a level equivalent to that of SIMC. This is a somewhat unfortunate situation for the IMC algorithm. Generally, we expect that computational results should improve as the time step is decreased. In IMC, the accuracy of results can become worse due to the increasing influence of teleportation error and the user must be very careful as a result.
The weight vector extension provides spectral information with a very good noise figure, especially for regions of frequency space that would be sampled with low probability in the line profile. Global results that are the result of frequency integrations do not benefit significantly from this extension (on scalar processors) due to the additional cost of computing exponentials for each element of the weight vector for each track the particle makes. Weight vectors have the advantage that the distance to the next zone has to be computed only once. On vector processors, employed in [5], where a vector of a dozen or so exponentials would take the same time as a single scalar exponential, the weight vector extension of SIMC does not require much additional time and improved algorithm performance is worthwhile even if spectral information is not required.
Geometric zoning, so that thin zones are used near transition regions, provides a significant improvement in the accuracy of the solution provided by SIMC. The length of tracks in the limit of large time step are controlled by the zoning in SIMC, and not by effective scattering. Careful zoning in SIMC, which inherently has a smaller execution time than IMC, improves the accuracy of the solution. Zones should be thin where the upper level atomic population varies rapidly in space. The strategy works well for SIMC if the error introduced by photon teleportation is carefully watched.
IMC can benefit from zone refinement, especially if small time steps are required to improve temporal resolution. However, a word of caution must be given here. The IMC algorithm can demonstrate poor convergence behavior due to teleportation error as the time step is reduced and we have actually seen results for IMC actually get worse when the time step is reduced independently of the zone size. IMC seems to be somewhat magical in that it delivers good results for coarse time steps and zoning, but its results can get worse in response to refinement if one is not careful. SIMC responds much more systematically to independent zone and time step refinements, with good convergence characteristics on both fronts.
Biasing significantly reduces the noise for SIMC, especially if the line center opacity is high. Biasing the spontaneous emission in favor of the thin zones near the surface of a transition region and in favor of thin subzones near the surface of a centrally located thick zone in cases of high line center opacity, can produce results of very high precision and low noise if one wants to examine the physics of a boundary layer. The IMC algorithm is less capable in this regard.
In general, all of the methods examined in this paper have their advantages and disadvantages. The teleportation issue provides an advantage for IMC, providing relatively coarse zoning and time step sizes are adequate. If one wants to perform significant zone and time step refinement in order to produce high accuracy results, however, SIMC becomes the method of choice with its better convergence behavior. If spectral information is required in a high opacity region or a transition region, the SIMC method extended with weight vectors provides the best method.
If the problem of teleportation were eliminated, the one disadvantage of SIMC evaporates, and the method would become the method of choice. We will examine the possibility of accomplishing this, using a new formulation for the transport equation, in future work.