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Astron. Astrophys. 342, 179-191 (1999)

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4. Summary and conclusions

We have derived a numerical approach which guarantees that the sum of mass fractions equals unity in simulations of multi-fluid flows with higher-order Godunov-type methods. Unlike other commonly used numerical methods, the proposed scheme preserves the conservative character of the underlying advection scheme. We would like to stress that even if the advection step is formally written in conservation form, this does not necessarily imply that the scheme is conservative in case of multi-fluid flows. This fact is often overlooked.

Modifications of the interpolation step are needed in higher-order Godunov-type methods to reduce the numerical diffusion near composition interfaces. These modifications are described together with procedures that ensure the monotonicity of the scheme.

The Consistent Multi-fluid Advection method (CMA) is proposed as a new method to accurately describe multi-fluid flows and is implemented in the PPM-based hydrodynamic code PROMETHEUS . Since the advection part of PPM is well tested for a single fluid, we have not considered simple test problems with known solutions, like e.g., the linear advection of square profiles in the mass fractions. Instead we investigated the behaviour of the CMA method in case of 1D test problems with both smooth and discontinuous composition profiles involving flows with strong hydrodynamic discontinuities. As for these problems no analytical solutions are known, the correctness of the proposed method has been demonstrated by means of a convergence study. Although other methods converge to the same solution too, the CMA method is the only one, which simultaneously guarantees the mass constraint [FORMULA], i.e. the sum of the mass fractions is always and everywhere equal to one.

In order to demonstrate the advantage of the CMA method, we have also studied a problem of astrophysical relevance, shock-induced thermonuclear burning in a supernova explosion. It is shown that numerical diffusion near composition interfaces can change the composition of supernova ejecta by a factor of a few. The abundance of titanium is most severely affected in our test calculations. The consequences of these findings for explosive nucleosynthesis calculations should be explored in more detail.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998