9.1 Incorporation of complex equations of state

Concerning the usage of complex equations of state, a limitation must be pointed out which hampers all numerical methods (HRSC, AV, symmetric schemes, etc.), and which is particularly troublesome for the Riemann solvers used in HRSC methods, even in the Newtonian limit. These problems are pronounced particularly in situations where phase transitions are encountered. Then the EOS may have a discontinuous adiabatic exponent and may even be non-convex. The Riemann solver of Colella and Glaz [59] often fails in these situations, because it is derived under the assumption of convexity of the EOS.

In elementary calculus, a function f (u) is convex if ∂2f (u)∕∂u2 ≥ 0 for all u. An EOS is called convex if its isentropes are convex in the p − V plane. Convexity of the isentropes is measured by the fundamental derivative (see, e.g., [191])

| 1V 2 ∂2p || 𝒢 = ---- ---2||, (71 ) 2γp ∂V S
where V = 1∕ρ is the specific volume, S the specific entropy, and γ = − ∂ logp∕ ∂ logV |S the adiabatic exponent, respectively. In particular, if 𝒢 > 0, then the isentropes are convex.

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., [154]).

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

2 || 2 Σ = ∂-p-||+ 2c21-−-cs, (72 ) ∂𝜖2 |σ s 𝜖 + p
where 2 cs ≡ ∂p∕∂𝜖|σ is the sound speed, and 𝜖,p, and σ are the energy density, pressure, and specific entropy, respectively [40Jump To The Next Citation Point]. For thermodynamically normal matter Σ > 0, and for so-called thermodynamically anomalous (TA) matter Σ < 0 [40]. 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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