STRUCTURE-PRESERVING DISCRETISATIONS FOR THE QUASI-INCOMPRESSIBLE, THERMAL MAXWELL–STEFAN EQUATIONS

Boris Andrews | Aaron Baier-Reinio | Aaron Brunk | Patrick Farrell | Mária Lukáčová-Medviďová

OCT.2026 (arXiv) | In review

We study a thermal Maxwell–Stefan–Navier–Stokes system modelling miscible mixtures under a quasi-incompressibility constraint. […] The first method […] guarantees generation of entropy and enforces the quasi-incompressibility constraint pointwise […]. The second method employs a continuous Petrov–Galerkin discretisation in time, which also generates entropy and satisfies the constraint pointwise, and in addition conserves the total energy and allows high-order time approximations. […]

FULL ABSTRACT

We study a thermal Maxwell–Stefan–Navier–Stokes system modelling miscible mixtures under a quasi-incompressibility constraint. The system consists of the conservation laws for mass and momentum, and the balance law for entropy, which is preferred over the balance law for energy because it facilitates the development of two structure-preserving discretisation methods.

The first method applies a semi-implicit Euler scheme in time, together with suitable finite elements in space, which guarantees generation of entropy and enforces the quasi-incompressibility constraint pointwise, but numerically dissipates total energy. The second method employs a continuous Petrov–Galerkin discretisation in time, which also generates entropy and satisfies the constraint pointwise, and in addition conserves the total energy and allows high-order time approximations. The structure-preserving properties and behaviour of both numerical methods are demonstrated by a series of numerical experiments.

(Further details available soon!)

CO-AUTHORS

Aaron Baier-Reinio

• Affiliation | University of Oxford
• Joint publications | Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell–Stefan equations

Aaron Brunk

• Affiliation | Johannes Gutenberg University Mainz
• Joint publications | Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell–Stefan equations

Patrick Farrell

• Affiliation | University of Oxford / Charles University
• Joint publications | Enforcing conservation laws and dissipation inequalities numerically via auxiliary variables / Helicity-preserving finite element discretization for magnetic relaxation / Conservative and dissipative discretisations of multi-conservative ODEs and GENERIC systems / Arbitrary-order structure-preserving discretizations for geometric curvature flows / Automated Galerkin time stepping in Irksome / Strongly enstrophy-stable integrators for the incompressible Navier–Stokes equations / Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell–Stefan equations / Energy-stable discretisations of viscoelastic flows*

Mária Lukáčová-Medviďová

• Affiliation | Johannes Gutenberg University Mainz
• Joint publications | Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell–Stefan equations