Higher-Order Entropy-Conservative Finite-Volume Methods

Definitions and mathematical statement

A finite-volume discretization evolves approximate cell averages by numerical fluxes across cell boundaries, with additional source contributions when the governing equation is a balance law. Entropy conservation refers to satisfying a discrete entropy balance without artificial entropy production; physical entropy-source contributions and boundary entropy fluxes must be accounted for when present. The governing equation, entropy, numerical entropy flux, and treatment of sources and boundaries are needed to specify this requirement mathematically.

If the reported construction has accuracy order $p_0$, a higher-order extension would have accuracy order $p>p_0$ in a specified consistency or convergence sense. Neither $p_0$, the requested orders $p$, nor the meaning of accuracy is supplied. In particular, the record does not specify whether higher order concerns space, time, or both.

The supplied research question is: Generalize the entropy-conservative finite-volume discretization to higher-order methods.

Here "the discretization" refers to a construction in the cited workshop report, but that construction is not identified or reproduced in the supplied record. The intended task is therefore to extend that construction to higher order while retaining its entropy-conservation property. A uniquely quantified mathematical statement cannot be recovered from the supplied evidence: the underlying equations, admissible data, entropy structure, numerical construction, accuracy requirements, boundary conditions, and scope of the requested generalization remain unspecified. No particular choices of these missing hypotheses are imposed in this draft.

Short English statement

Extend the finite-volume construction referenced by the source to higher accuracy while preserving its discrete entropy balance. The supplied record does not identify the construction or specify the equations, accuracy orders, or boundary and source treatment, so a precise quantified target requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004642$, title Higher-Order Entropy-Conservative Finite-Volume Methods, source problem code catalogue entry. The record gives proposed year 2021 and names no proposer. Its statement-status field is exact; its verification field says the wording was retained byte-for-byte and required no statement-level reconstruction. These are supplied registry qualifications.
• Supplied source citation: Hyperbolic Balance Laws: modeling, analysis, and numerics (2022), Oberwolfach Reports, pp. 589--661, DOI 10.4171/owr/2021/11. Supplied source URL: https://doi.org/10.4171/owr/2021/11. The background attributes the collection to Oberwolfach workshop reports and their open-problem discussions; no individual contribution or problem page is identified.
• Supplied literature reference: Oberwolfach Report 11/2021, https://doi.org/10.4171/owr/2021/11. Literature checked at: 2026-08-22. This is supplied dated curation, not an independently verified assessment. The registry labels the problem open and states that higher-order extensions remain unresolved in the reported generality, despite convergence and entropy results for selected lower-order schemes and systems. Its background explicitly requests independent verification of time-sensitive status and cited claims.