Energy/AP audit and a kinetic-cone counterexample for a stiff two-moment model

Definitions and mathematical statement

This record is Bruno Després's participant-interest note, not a stated open problem. It discusses finite-volume convergence via consistency plus stability, including nonlinear scalar conservation laws, and work with Christophe Buet on implicit approximations of a stiff two-moment model. The first extracted equation is $u_t+\varepsilon^{-1}v_x=0$; the second lacks a spatial derivative on its flux. The supplied verification explicitly reconstructs \[ u_t+\varepsilon^{-1}v_x=0,\qquad v_t+\varepsilon^{-1}\partial_x f(u,v)=-\sigma\varepsilon^{-2}v. \] This derivative is an inference, not verified original typography. The flux, boundary conditions, and detailed original numerical solver are not supplied. The participant reports an implicit one-dimensional solver using a linear system with frozen coefficients, preserving the kinetic constraint $|v|/u\le1$ and having the correct diffusion limit. In nonnegative variables the closed kinetic cone is $\mathcal K=\{(u,v):u\ge|v|\}$; the displayed ratio itself presupposes $u>0$.

The aid studies the explicitly reconstructed linear telegraph specialization $f(u,v)=u$, $\sigma>0$, $\varepsilon>0$, rather than identifying the participant's nonlinear solver. Its formal diffusion limit is $u_t=\sigma^{-1}u_{xx}$. It reports an exact $\varepsilon$-uniform discrete $L^2$ energy identity for a fully implicit staggered backward-Euler discretization, but the stencil, boundary conditions, and identity are not present in the supplied record.

Its asymptotic-preserving claim is specifically a fixed-grid, fixed-time-step limit: if the initial grid vector $u_\varepsilon^0\to\bar u^0$ and $v_\varepsilon^0$ remains uniformly bounded as $\varepsilon\downarrow0$, then at each fixed time index the computed $u_\varepsilon^n$ converges to the standard three-point backward-Euler diffusion sequence initialized by $\bar u^0$. On a uniform grid with spacing $h$ and step $\tau$, that limiting interior update is \[ \frac{\bar u_j^{n+1}-\bar u_j^n}{\tau} =\frac{\bar u_{j+1}^{n+1}-2\bar u_j^{n+1}+\bar u_{j-1}^{n+1}}{\sigma h^2}, \] with compatible boundary conditions still to be specified. No initial requirement $v_\varepsilon^0=O(\varepsilon)$ is imposed in the reported result. This is not a simultaneous continuum-limit or uniform-in-time convergence assertion.

The aid separately reports an exact four-cell periodic counterexample for the collocated centered implicit analogue: cone-admissible initial cell states produce an updated cell with $u<|v|$. The update parameters and rational cell values are not included here. This distinguishes discrete energy stability and a correct fixed-grid diffusion limit from preservation of $\mathcal K$; it does not refute the participant's separate claim for an unspecified different solver.

Short English statement

This is a participant note about implicit relaxation solvers. Its damaged flux is explicitly reconstructed, and the aid audits a linear telegraph specialization. The reported energy identity and fixed-grid diffusion limit do not by themselves imply preservation of the kinetic cone.

Sources

• Local UnsolvedMath registry, permanent ID 20002457, source code catalogue entry; created and updated 14 August 2026. Canonical aim-pdes-notes.json, zero-based notes[68], tag section, item A.4, "Despres, Bruno."
• AIM workshop "Stiff sources and numerical methods for conservation laws": https://aimath.org/WWN/balancelaws/balancelaws.pdf. The note names collaborator Christophe Buet. Original remarks and literature field are empty; the statement is marked reconstructed_unverified.
• Supplied verification refers to later two-moment radiation equations of Buet and Després as motivation for restoring the flux derivative, but gives no complete citation. Research attempt 1: context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low. Its advertised exact stencil, identity, and rational counterexample are not included in this source record.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.