We use three test problems to probe the properties of the implicit Monte Carlo methods studied in this paper. The first example is a simple collisionally pumped system used in Refs. [3] and [4], although we push to a much higher opacity given the increased speed of modern microprocessors. The second example models the heating of cold media by incident radiation. The third example, which has an analytical solution available for comparison purposes, involves collisional pumping in the center of the slab.
In comparing the methods, the two issues of concern are systematic and statistical errors. The systematic error is an error in the numerical modeling of the physics that persists in the limit of large Monte Carlo particle count. It can be controlled by suitably refining the time step size, the choice of zones, the choice for frequency bins, or by modifying a given method to be higher order accurate in these parameters. The implicit transport methods examined in this paper develop subtle interplay between these different controls on discretization error, and we will demonstrate this in the results presented below.
Given the discretization parameters that control systematic error (and any importance sampling scheme) the number of Monte Carlo particles controls the statistical error for a given problem run. In the limit of large particle count, the statistical error scales inversely with the square root of the particle count. This fact provides a means to evaluate the relative efficiency of the methods. We choose a particle count for each method that results in a given execution time (first confirming that the noise envelope, in fact the standard deviation of individual runs, is scaling like the square root of the particle count in each case) and then examine the noise envelope for 100 independent runs. The relative speeds of the methods, in the limit of large particle count, are then given as the square of the ratios of the measured noise envelopes. We would like to note that while this is a good way to compare the relative efficiency of the methods under examination, we do not suggest that this is how users should employ Monte Carlo applications in practice.
Unless otherwise stated, each example will use 16 frequency groups, each of 0.2 Doppler widths. In addition, the line profile, and therefore the frequency spectrum, is symmetric around zero and we take advantage of this symmetry. Spatially, each problem is divided up into 21 equally spaced zones unless noted otherwise.
In the results below, problem output is always presented as the mean and standard deviation (not error in the mean) of 100 independent problem runs. The mean of a large number of problem runs provides the best opportunity to spot systematic error, while the standard deviation gives us a good idea of just how much scatter would be present in a single run along with a reliable way to estimate the computational efficiency of the method.