One of the most important factors determining the quality of a numerical algorithm is its robustness. In the simplest case it can be regarded as a property of the scheme to provide the result at minimum cost once the accuracy is specified. In numerical hydrodynamics much effort has been spent during the last two decades on improving advection schemes in such a way that they do not only provide high accuracy in regions of smooth flows but also resolve flow discontinuities (shocks and contact discontinuities) with a minimum number of discrete grid zones. In the past a variety of numerical experiments were conducted to compare the overall quality of solutions obtained with the help of different advection schemes which provided an understanding of the advantages and disadvantages of already available or newly proposed algorithms (Colella & Woodward 1984, Carpenter et al. 1990, Fryxell et al. 1991, Yang & Przekwas 1992, Stone & Norman 1992, Steinmetz & Müller 1993, Kang et al. 1994).
For our numerical experiments we have used the Piecewise Parabolic Method (PPM) of Colella & Woodward 1984; hereafter CW) to study the evolution of multi-fluid flows with strong discontinuities and stiff source terms. In theoretical astrophysics the PPM method has been used to study a range of hydrodynamic phenomena like stellar collisions (Ruffert & Müller 1990, Frolov et al. 1994), evolution of supersonic jets (Balsara & Norman 1992, Basset & Woodward 1995), large-scale structure formation in cosmology (Bryan et al. 1994), interaction of stellar winds in massive close binaries (Stevens et al. 1992), and the stability of radiative shock waves (Strickland & Blondin 1995). Numerical experiments of CW and Yang & Przekwas (1992) clearly demonstrated the superiority of the PPM scheme among several modern advection schemes.
A particularly interesting and challenging astrophysical problem involving multi-fluid flow (and one of our numerical experiments; see Sect. 3.4) is the simulation of mixing in supernova envelopes. Mixing occurs because the non-steady propagation of the shock wave formed after core collapse gives rise to Rayleigh-Taylor instabilities (for a recent review, see e.g., Müller 1998). The first (2D) simulations of mixing involving ten separate fluids were performed by Arnett et al. (1989). They computed the propagation of the supernova shock wave, which was artificially created by a "point" explosion, through the envelope of a realistic stellar model. Better resolved and more detailed simulations were later performed by Fryxell et al. (1991) and Müller et al. (1991). They identified the Rayleigh-Taylor unstable regions as being associated with discontinuities in the chemical composition in the envelope of the progenitor star. The simulations were performed with the PPM-based hydrodynamic code PROMETHEUS , which keeps track of different nuclear species by solving a set of additional continuity equations (see Fryxell et al. 1989).
In the case of single-fluid advection the problem of diffusion across contact discontinuities plays a crucial role and provides a simple test case for studying mixing between different fluids in numerical simulations. In multi-fluid flows both chemical and contact discontinuities may be present. Although, as we shall see later, there exist important differences between both kinds of discontinuities, we nevertheless can profit from our experience of modeling contact discontinuities when dealing with composition discontinuities. In this context we point out that Fryxell et al. (1989) demonstrated that the advection of a contact discontinuity is simulated better with a PPM scheme than with any other scheme they considered in their study.
Mixing of different fluids cannot be ignored if the chemical composition plays an important role in the hydrodynamic evolution. For example, in case of a realistic equation of state the total gas pressure is calculated as the sum of partial pressures exerted by each kind of species. More complex physical processes, emission from an optically thin medium (radiative cooling) or absorption of radiation, are strongly sensitive to changes in chemical composition, especially to changes in the heavy element abundances. Last but not least, the process of nuclear burning, to which we will pay special attention later in this paper, directly depends on the amount and type of nuclear fuel. In this particular case, mixing of different nuclear species due to numerical diffusion can substantially affect the final chemical composition or even the overall dynamics of the flow (for a recent review, see e.g., Müller 1998).
The paper is organized as follows. In Sect. 2 we briefly describe the basic components of the PPM scheme and some specific features implemented into the PROMETHEUS version used in our numerical experiments. We then give a detailed description of the new consistent multi-fluid advection method. In Sect. 3 we present the results of our test simulations. A discussion of the results is contained in Sect. 4.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998