A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows
Abstract
We consider the fluid subsystem of a variationally derived hyperbolic thermodynamically compatible (HTC) model of compressible multiphase flows. Its total energy is in general not convex in the natural state variables, so a scalar numerical viscosity acting on them need not produce non-negative entropy. Reformulating the equations in terms of the volume fractions separates changes in the amount of each phase from changes in its intrinsic state. This guides the construction of a configurational numerical viscosity in phase variables, in which the energy is convex, while the scheme retains the structure of the HTC model. At the semi-discrete level, the new schemes satisfy: I) local conservation of the phase masses, the mixture momentum and the total energy; II) a cell entropy inequality whose production is non-negative face by face and phase by phase, with no convexity assumption in the evolved variables and no clipping; III) positivity of the partial densities and volume fractions under explicit time-step conditions; IV) exact preservation of contact equilibria; V) equivalence with a scheme for the density-weighted volume fractions. The key ingredients are: i) an exact decomposition of the jump of each phase across a face into a configurational part, dissipated with the material speed, and an intrinsic part, dissipated with the acoustic speed; ii) an exchange form of this dissipation between neighbouring cells, with non-negative weights; iii) a side-wise transfer of the dissipation to the evolved variables through the chain rule of each cell, which turns the energy defects into Bregman divergences of the single-fluid energies; iv) a thermodynamic correction of the reversible part built in the phase variables. Properties I)-V) are proved and verified numerically, including a shock tube in which the Rusanov viscosity produces negative entropy and a shock-water column interaction.
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BibTeX
@misc{ferrari2026configurational,
title = {{A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows}},
author = {Davide Ferrari and Ilya Peshkov and Michael Dumbser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23247},
eprint = {2610.23247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23247},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows
AU - Davide Ferrari
AU - Ilya Peshkov
AU - Michael Dumbser
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23247
UR - https://arxiv.org/abs/2610.23247
AB - We consider the fluid subsystem of a variationally derived hyperbolic thermodynamically compatible (HTC) model of compressible multiphase flows. Its total energy is in general not convex in the natural state variables, so a scalar numerical viscosity acting on them need not produce non-negative entropy. Reformulating the equations in terms of the volume fractions separates changes in the amount of each phase from changes in its intrinsic state. This guides the construction of a configurational numerical viscosity in phase variables, in which the energy is convex, while the scheme retains the structure of the HTC model. At the semi-discrete level, the new schemes satisfy: I) local conservation of the phase masses, the mixture momentum and the total energy; II) a cell entropy inequality whose production is non-negative face by face and phase by phase, with no convexity assumption in the evolved variables and no clipping; III) positivity of the partial densities and volume fractions under explicit time-step conditions; IV) exact preservation of contact equilibria; V) equivalence with a scheme for the density-weighted volume fractions. The key ingredients are: i) an exact decomposition of the jump of each phase across a face into a configurational part, dissipated with the material speed, and an intrinsic part, dissipated with the acoustic speed; ii) an exchange form of this dissipation between neighbouring cells, with non-negative weights; iii) a side-wise transfer of the dissipation to the evolved variables through the chain rule of each cell, which turns the energy defects into Bregman divergences of the single-fluid energies; iv) a thermodynamic correction of the reversible part built in the phase variables. Properties I)-V) are proved and verified numerically, including a shock tube in which the Rusanov viscosity produces negative entropy and a shock-water column interaction.
ER -