Analysis of Reaction-Diffusion Systems for Flame Capturing in Type Ia Supernova Simulations

We present a study of numerical behavior of a thickened flame used in Flame Capturing (FC, Khokhlov (1995)) for tracking thin unresolved physical flames in deflagration simulations. We develop a steady-state procedure for calibrating the flame model used, and test it against analytical results. We observe numerical noises generated by original realization of the technique. Alternative artificial burning rates are discussed, which produce acceptably quiet flames. Two new quiet models are calibrated to yield required "flame" speed and width, and further studied in 2D and 3D setting. Landau-Darrieus type instabilities of the flames are observed. One model also shows significantly anisotropic propagation speed on the grid, both effects increasingly pronounced at larger matter expansion as a result of burning; this makes the model unacceptable for use in type Ia supernova simulations. Another model looks promising for use in flame capturing at fuel to ash density ratio of order 3 and below. That "Model B" yields flames completely localized within a region 6 cells wide at any expansion. We study Markstein effect for flame models described, through direct numerical simulations and through quasi-steady technique developed. Comparison demonstrates that Markstein effect dominates instability effects on curved flame speed for Model B in 2D simulations for fuel to ash density ratio of about 2.5 and below. We find critical wavelength of LD instability by direct simulations of perturbed nearly planar flames; these agree with analytical estimates when Markstein number values found in this work are used. We conclude that the behavior of model B is well understood, and optimal for FC applications among all flame models proposed to date.