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I use MESA right now to play around with stellar dynamics, but it can't do mass ejections and therefore, I can't think of a way to let the simulated stars explode as supernovae.

What simulation software would I use for these?

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Regarding Terminology:
'Mass ejections' (at least semantically) are very different from supernovae. A "mass ejection" would generally refer to something like what the sun does as part of its normal activity --- ejecting very small amounts of plasma, or on the other end of the spectrum, the ejection of massive shells of material by very massive (e.g. $M \gtrsim 40 M_\odot$) stars$^{[1]}$. Supernovae, on the other hand, are the explosion and destruction of an entire star (or at least everything outside of the core).

Simulations
The sad fact of the matter is that there are no simulations which can fully self-consistently explode a star. In other words, there is no code that can start with a star and have it naturally explode$^{[2]}$. There are many specialized codes, however, which deal with particular aspects of the explosions. The initial collapse and explosion is usually done with very specialized, complex hydrodynamic codes by people like Christian Ott, Adam Burrows, Chris Fryer, etc. These codes are not public.

More general hydrodynamic codes are usually used to study the effects of supernovae, i.e. how the blast waves evolve under different circumstances. In these simulations, the user artificially deposits a huge amount of energy in the center of a stellar model (mimicking the energy produced by an explosion), and then sees how the system evolves. One of the most popular and advanced codes for this is called FLASH, produced and managed at the U of Chicago.


[1] For a technical article see, e.g.: http://arxiv.org/abs/1010.3718
[2] There are numerous reasons for this, first and foremost that it is an incredibly difficult computational task. Stars are generally about $10^6$ km in size, while their cores are only a few km --- to understand the explosion, the code needs to model this entire dynamic range which is incredibly challening. Additionally, the densities and temperatures involved in the cores of supernovae are way outside of the range of whats ever been explored in a laboratory --- so information about that materials behavior is still largely uncertain. There are lots of other big challenges (e.g. incorporating general relativity, advanced nuclear reactions, etc) but these are some of the key issues people are exploring.

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