Convergence Without Continuous or TVD Numerical Fluxes

Definitions and mathematical statement

A finite-volume scheme evolves cell averages using numerical fluxes across cell boundaries. A total-variation-diminishing (TVD) condition requires that the discrete spatial total variation not increase under a time step, subject to whatever mesh and time-step restrictions are imposed. The supplied record does not specify the relevant discrete total variation, numerical flux continuity condition, or admissible restrictions.

Source question. Can a finite-volume scheme whose numerical flux satisfies neither the usual continuity condition nor a TVD condition nevertheless converge, particularly in the setting of VOF--machine-learning solvers?

Unresolved formulation. This is an incomplete research question. The governing equation, unknowns and their domains, initial and boundary data, numerical update, refinement procedure, convergence topology, and intended limiting solution are unspecified. The phrase "usual continuity condition" has no supplied definition. It is also unclear whether the question requests an example, a convergence criterion for a specified class, or a general convergence result. Consequently, no precise quantified mathematical assertion can be recovered without additional source information.

Short English statement

The supplied question asks whether finite-volume methods can converge when their numerical fluxes satisfy neither the intended continuity requirement nor the intended TVD requirement, with particular interest in VOF--machine-learning solvers. The hypotheses and meaning of convergence remain unspecified, so this draft preserves the question and its unresolved qualifications.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004641$, source problem code catalogue entry, entitled "Convergence Without Continuous or TVD Numerical Fluxes." The record belongs to the Oberwolfach Reports Open Problems collection and supplies proposed year 2021; no individual proposer is named.
• Supplied report citation: Hyperbolic Balance Laws: modeling, analysis, and numerics (2022), Oberwolfach Reports, pp. 589--661, DOI 10.4171/owr/2021/11. Supplied source and literature reference, labeled "Oberwolfach Report 11/2021": https://doi.org/10.4171/owr/2021/11. The background describes these reports as records of mathematical workshops and their open-problem discussions. No problem-specific page or complete source passage is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, while reporting that the extracted record contains only the words "An open problem" and a consequence, insufficient to verify a complete mathematical claim. This is supplied dated curation, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry also supplies statement status corrected_verified, explained as a publication-format audit preserving mathematical content while repairing notation or typography. This supplied label does not resolve the missing mathematical hypotheses or the extraction limitation reported by the dated curation.