Layer-wise discrete certificates in multicomponent reacting-flow discretizations
Abstract
Multicomponent reacting-flow discretizations must conserve total species mass, preserve positive mass fractions, and dissipate a convex mixing potential. These properties are controlled at different stages of a solver, however, and a certificate established at one stage need not survive the next. We study this compatibility problem for mixture-averaged diffusion, non-orthogonal finite-volume reconstruction, and positivity-preserving Patankar time integration. The analysis is isothermal, uses prescribed density, and uses synthetic detailed-balance reaction networks for the chemistry tests. At the face level, the entropy-variable jump is shown to equal a log-mean Gibbs Hessian applied exactly to the mass-fraction jump. Requiring conservation and entropy neutrality of the mass-correction term on the simplex tangent space then yields a unique componentwise-positive face composition and an exact two-point entropy certificate. The arithmetic-weight face criterion is also satisfied by production neutral-air/carbon data and a quasi-neutral ambipolar-air control, but a face-level certificate does not automatically extend to subsequent discretization layers. A weighted least-squares non-orthogonal correction can, in the tested prototype, convert a dissipative two-point operator into an anti-dissipative multipoint operator. Likewise, Patankar rescaling can preserve positivity and conservation while increasing the free energy; an explicit finite-step rational counterexample establishes this failure independently of ensemble statistics. A fully discrete two-cell example further shows that a positive face entropy margin does not guarantee positivity of the fully discrete update. Thus conservation, positivity, and free-energy dissipation are layer-specific properties and require distinct discrete certificates.
Related
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BibTeX
@misc{zhang2026layerwise,
title = {{Layer-wise discrete certificates in multicomponent reacting-flow discretizations}},
author = {Jingchao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27272},
eprint = {2609.27272},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27272},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Layer-wise discrete certificates in multicomponent reacting-flow discretizations
AU - Jingchao Zhang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27272
UR - https://arxiv.org/abs/2609.27272
AB - Multicomponent reacting-flow discretizations must conserve total species mass, preserve positive mass fractions, and dissipate a convex mixing potential. These properties are controlled at different stages of a solver, however, and a certificate established at one stage need not survive the next. We study this compatibility problem for mixture-averaged diffusion, non-orthogonal finite-volume reconstruction, and positivity-preserving Patankar time integration. The analysis is isothermal, uses prescribed density, and uses synthetic detailed-balance reaction networks for the chemistry tests. At the face level, the entropy-variable jump is shown to equal a log-mean Gibbs Hessian applied exactly to the mass-fraction jump. Requiring conservation and entropy neutrality of the mass-correction term on the simplex tangent space then yields a unique componentwise-positive face composition and an exact two-point entropy certificate. The arithmetic-weight face criterion is also satisfied by production neutral-air/carbon data and a quasi-neutral ambipolar-air control, but a face-level certificate does not automatically extend to subsequent discretization layers. A weighted least-squares non-orthogonal correction can, in the tested prototype, convert a dissipative two-point operator into an anti-dissipative multipoint operator. Likewise, Patankar rescaling can preserve positivity and conservation while increasing the free energy; an explicit finite-step rational counterexample establishes this failure independently of ensemble statistics. A fully discrete two-cell example further shows that a positive face entropy margin does not guarantee positivity of the fully discrete update. Thus conservation, positivity, and free-energy dissipation are layer-specific properties and require distinct discrete certificates.
ER -