Spectrum formation in supernovae: Numerical techniques

We have combined several novel techniques for spectrum simulation in the computer program EDDINGTON which solves the comoving frame equation of transfer coupled with the statistical and radiative equilibrium equations. The first of these is a generalization of the accelerated lambda iteration (ALI) scheme to include an approximate frequency-derivative operator. This greatly enhances the convergence rate of ALI in optically thick, high-velocity shear flows. The next is a partial linearization technique which is capable of efficiently solving a very large (∼10⁴) number of rate equations on a moderately sized computer; part of its efficiency derives from a "fixed-excitation" iteration which allows this technique to handle simulations with a large number of (intrinsically) overlapping lines and continua. Finally, we derive an expansion opacity and emissivity approximation which allows us to determine the effect on the transfer and statistical equilibrium of a very large number of lines not explicitly represented in the frequency grid and additionally to treat line-blanketing from species not explicitly included in the rate equations. We illustrate the utility of these techniques with models of two supernovae. The first is a typical Type II supernova 45 days past explosion which illustrates the power of the ALI scheme for optically thick problems in rapidly moving flows. The second is a Type Ia supernova 250 days past explosion which demonstrates the ability of partial linearization and the expansion opacity/emissivity approximation to treat a problem with 727 atomic energy levels coupled by all continua and 4447 lines. For each we discuss rates of convergence and the effect of various convergence-accelerating techniques. Detailed models of various supernovae and the microphysics (e.g., energy deposition and atomic data) we employ will be discussed in future publications.