Tadmor's framework: design numerical fluxes for conservation laws that exactly preserve entropy at the discrete level. The framework works beautifully for ideal gases, where entropy is convex and the theory is clean. Real gases — supercritical CO₂, transcritical fuel injection, non-ideal equations of state — break the assumptions.
The paper generalizes the framework to secondary structures that may be non-convex and have singular Hessians. A secondary structure is any quantity (kinetic energy, entropy, a non-convex thermodynamic potential) that satisfies its own balance law as a consequence of the primary conservation laws. When this secondary structure is convex, Tadmor's original conditions apply. When it's non-convex, the Hessian has a kernel, and the standard approach produces an ill-posed system.
The fix: null-consistency. This is a structural requirement that the numerical flux must satisfy in the kernel of the Hessian — the directions where the secondary structure's curvature vanishes. Without null-consistency, the discrete balance law for the secondary structure can't be enforced well-posedly. With it, the discrete gradient construction extends to arbitrary equations of state.
The practical result: an entropy-conserving, kinetic-energy-consistent flux for the Euler equations with any equation of state. Validated on supercritical nitrogen, transcritical mixing layers, and fully turbulent transcritical flows — regimes where the equation of state passes through phase boundaries and the Hessian of the secondary structure genuinely degenerates.
The null space of the Hessian is not an obstacle but a design constraint. The flux must be consistent precisely where the curvature information is absent.