Entropy-Compatible Barrier Schemes for Diffusive FENE Flows

Abstract: FENE-type conformation-tensor models impose a finite-extensibility constraint that is absent from Oldroyd--B flow: the conformation tensor must satisfy $\CC\succ0$ and $\tr\CC<L^2$. Positive definiteness alone is therefore insufficient, since a numerical state can remain positive while crossing the singular trace barrier. Even a trace-preserving logarithmic parametrization is not enough by itself: high-order reconstruction can remain inside the finite-extensibility domain while injecting artificial FENE entropy. We develop and analyze a barrier-preserving entropy-compatible discretization for FENE-P type flows with polymer center-of-mass molecular diffusion and for trace-singular FENE-family closures with the same entropy structure. The method combines a trace-barrier free energy, a finite-extensibility logarithmic parametrization, a least-damping entropy-compatible barrier-log reconstruction, molecular diffusion paired with the barrier entropy variable, compatible quadrature for polymeric work, and a scaled FENE stress variable for the small-Weissenberg limit. For admissible discrete states we prove finite-extensibility preservation at entropy quadrature points, existence and bisection computability of the maximal entropy-admissible reconstruction parameter, a fully discrete free-energy inequality with relaxation and molecular-diffusion barrier dissipation, a quantitative AP stress closure, and a fixed-discretization Newtonian limit. A conditional relative-entropy estimate is derived on compact subsets of the finite-extensibility domain. Numerical diagnostics verify barrier preservation, entropy-compatible reconstruction, energy decay, AP closure, coupled velocity--pressure--stress accuracy, and high-Weissenberg robustness near the trace constraint.