In elementary calculus, a function is convex if for all . An EOS is called convex if its isentropes are convex in the plane. Convexity of the isentropes is measured by the fundamental derivative (see, e.g., )
For a perfect (ideal) gas, a jump discontinuity in the initial conditions of the hydrodynamic equations gives rise to at most one (compressional) shock, one contact, and one simple centered expansion fan, i.e., one wave per conservation equation. For a real gas, however, the EOS can be nonconvex. If that is the case, the character of the solution to the Riemann problem changes, resulting in anomalous wave structures. In particular, the solution may be no longer unique, i.e., a jump discontinuity in the initial conditions may give rise to multiple shocks, multiple contacts, and multiple simple centered expansion fans (see, e.g., ).
Situations where phase transitions cause a discontinous adiabatic index or non-convexity of the EOS are encountered, e.g., in simulations of neutron star formation, simulations of the early Universe, and simulations of relativistic heavy ion collisions (see Section 7.3).
Matter undergoing a first-order phase transition may exhibit thermodynamically anomalous behaviour signalled by a change of sign of the quantity. For thermodynamically normal matter , and for so-called thermodynamically anomalous (TA) matter . For the expansion (compression) of thermodynamically normal matter a rarefaction wave (a compressional shock) is the stable hydrodynamic solution, while it is a rarefaction shock (compressional simple wave) in case of TA matter.
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