I am having the conceptual doubt. I am reading in many papers that HLLC is positivity preserving but Roe scheme is not. But I don't know, how to understand why is it so?
If anyone can kindly explain to me in detail, it will really be a good help for me.
A good question and interesting, but its answer can be long and involved. The short answer as to why the classical Roe flux is not positivity preserving deals with how the flux is derived from the linearised Riemann problem. In short, there are too few degrees of freedom to impose positivity conditions along with consistency. See the article "Low dissipation entropy fix for positivity preserving Roe's scheme" by M. Pelanti et al. That is the shortest answer that I can offer. Otherwise, for more technical discussion, see "ON THE CHOICE OF WAVE SPEEDS FOR THE HLLC RIEMANN SOLVER", SIAM J. SCI. COMPUT., Vol. 18, No. 6, pp. 1553–1570, November 1997 by Batten et al. The math shown here will take you through the arguments that show that HLLC is positivity preserving. It takes time, and there are many inequalities that you must sort through. The core paper that discusses why linearised Riemann solvers cannot be positivity preserving (without fixes) is by B. Einfeldt, "On Godunov-type methods near low densities", J. Comput. Phys., 92 (1991), pp. 273–295. Hopefully, these articles will help answer your question.
Thanks a lot for your answer and references. Yes it is indeed long and involved but worth reading.
Thank you.
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