Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell-Stefan equations

Boris D. Andrews; Aaron Baier-Reinio; Aaron Brunk; Patrick E. Farrell; Mária Lukáčová-Medviďová

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Structure-preserving discretisations for the quasi-incompressible, thermal Maxwell-Stefan equations

Boris D. Andrews, Aaron Baier-Reinio, Aaron Brunk, Patrick E. Farrell, Mária Lukáčová-Medviďová

arXiv:2610.35157v1Oct 26, 2026math.NAcs.NA

TL;DR

The paper studies convection and diffusion in a single-phase mixture of N > 1 chemical species, with barycentric momentum balance and specific internal-energy evolution. The authors replace the energy balance with an entropy balance and develop semi-implicit Euler and continuous Petrov–Galerkin finite-element schemes. Both schemes generate entropy and enforce quasi-incompressibility pointwise; semi-implicit Euler dissipates total energy, while CPG conserves it and supports arbitrary-order time approximations.

Problem
The paper studies convection and diffusion in a single-phase mixture of N > 1 chemical species, with barycentric momentum balance and specific internal-energy evolution.
Source: §1 · p. 2
Method
The authors replace the energy balance with an entropy balance and develop semi-implicit Euler and continuous Petrov–Galerkin finite-element schemes.
Source: §1 · p. 2, §1.3 · p. 5
Results
Both schemes generate entropy and enforce quasi-incompressibility pointwise; semi-implicit Euler dissipates total energy, while CPG conserves it and supports arbitrary-order time approximations.
Source: §1.3 · p. 5
Takeaways & Limitations
The two schemes offer different structure-preserving trade-offs: the simpler, cheaper semi-implicit method dissipates energy, whereas the more expensive CPG method conserves it.
Source: §1.3 · p. 5
Exact energy conservation in the CPG proof requires exact evaluation of specified integrals, while practical quadrature leaves energy error below that induced by inexact nonlinear solves.
Source: §5.2 · p. 21

Abstract

from arXiv · show

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.

1. Introduction

The paper develops a thermal Maxwell–Stefan–Navier–Stokes model for miscible mixtures subject to quasi-incompressibility, then designs finite-element schemes that preserve selected physical structures. The schemes differ in their treatment of total energy, and the work places them in a literature where preserving energetic or entropic structures for coupled Maxwell–Stefan–Navier–Stokes flows has been uncommon.

1. Introduction

The model couples species mass transport by convection and diffusion with barycentric momentum and thermal evolution, under a quasi-incompressibility constraint.The constraint, enforced through pressure, characterizes div u through the diffusion fluxes.Source: §1 · p. 2, §1.1 · p. 2

1.2. State equations.

Thermodynamic state relations connect internal energy, Helmholtz free energy, entropy, temperature, chemical potentials, and the pressure multiplier.The paper gives the Georgiadis–Jüngel mixing energy as a concrete example and models fluxes through thermodynamic driving forces.Source: §1.2 · pp. 2–3
The flux formulation incorporates Maxwell–Stefan diffusion and thermal cross-effects, while the stress combines an isotropic part with viscous dissipation.The stated assumptions include a symmetric positive-semidefinite Onsager matrix and positive heat conductivity.Source: §1.2 · pp. 3–4

1.3. Main results.

Both finite-element schemes conserve partial masses, generate entropy, and enforce quasi-incompressibility pointwise; semi-implicit Euler dissipates total energy, whereas CPG conserves it and extends to arbitrary order.The CPG scheme is more complicated and expensive than the first scheme.Source: §1.3 · p. 5

1.4. State of the art.

For thermal Maxwell–Stefan systems coupled to Navier–Stokes flow, the authors report that prior numerical methods generally did not preserve energetic or entropic structures, framing this work as an extension of.They also state that they are unaware of analytical treatment of the thermal system considered here.Source: §1.4 · pp. 5–6

2. Notation and preliminaries

The preliminaries specify assumptions used for the analysis and introduce a bound on changes in internal energy density.

Notation and preliminaries

The viscous dissipation is coercive in H1(Ω), modulo the symmetric-gradient kernel, when viscosity is uniformly bounded below and dλ + η ≥ 0.The setup also assumes finite-energy initial data satisfying the quasi-incompressibility constraint.Source: §2 · p. 6
A bound on changes in internal energy density is established assuming the Helmholtz free energy is convex in densities at fixed temperature and concave in temperature at fixed densities.The bound is intended for analysis of the semi-implicit Euler scheme.Source: §2 · p. 6

3. Entropy balance and global structures

The paper replaces the internal-energy balance with an entropy balance to facilitate numerical approximation, then derives global identities for smooth solutions.

Entropy balance and global structures

For smooth solutions, the internal-energy equation can equivalently be replaced by the total-energy or entropy equation; the system is reformulated using entropy to facilitate numerical approximation, as in.Source: §3 · p. 7
The entropy balance includes an entropy flux and an entropy production rate.The passage introduces these quantities as part of the balance law.Source: §3 · p. 7
The global structures for smooth solutions include conservation of partial masses and total energy, together with non-negativity of entropy production.The proof uses the mass, momentum, and energy equations and integration over the domain.Source: §3 · p. 8

4. Variational formulations

The paper constructs two variational formulations using the entropy balance to support discretisations with different structural properties. The first suits semi-implicit Euler, while the second is designed so its test functions remain valid under continuous Petrov–Galerkin discretisation.

4.1. Formulation I.

The first variational formulation replaces the energy balance with entropy balance and uses integration-by-parts identities to express the momentum equation weakly.Its structural identities follow from selected simple test functions.Source: §4 · p. 8, §4.1 · p. 9
The first formulation encodes mass conservation, entropy generation, and energy structure through simple test functions, including velocity u and temperature ϑ.The resulting identities are obtained by combining the tested equations so their right-hand-side terms cancel.Source: §4.1 · pp. 10–11
The first formulation supports semi-implicit Euler with entropy generation and quasi-incompressibility enforcement, but the scheme dissipates total energy.The energy is conserved at the continuous level.Source: §4.1 · p. 11
For certain nearly incompressible mixtures, thermodynamic forcing can dominate momentum; removing it when the free energy is degree-one homogeneous causes energy-stability results to fail for both schemes.For the benzene–cyclohexane example, the Lagrange number is approximately 10^15.Source: §4.1 · p. 12

4.2. Formulation II.

The second formulation introduces weighted velocity m = √ϱu and internal entropy density ϱs, with temperature and entropy density related implicitly through the thermodynamic relation.It uses the internal-energy density in the chemical-potential equation.Source: §4.2 · pp. 12–13
Unlike the first formulation, the second keeps its structural test functions valid after CPG discretisation in time.The formulation encodes the structures through simple test functions.Source: §4.2 · p. 13

5. Discretisation and analysis

This section specifies spatial finite element choices and analyzes two time discretisations. Both preserve entropy generation and pointwise quasi-incompressibility, while differing in energy behavior, cost, and time-order flexibility.

5. Discretisation and analysis

The spatial discretisation uses meshes of simplices and requires inf-sup stability and H1-conformity; the authors use Taylor–Hood spaces while noting compatible alternatives.Source: §5 · pp. 14–15
For polynomial degree ℓ ≥ 2, the chosen space cannot guarantee positivity of discrete temperatures and partial densities, so cutoff or regularisation may be needed to prevent solver divergence.Source: §5 · p. 15

5.1. Scheme I.

Under the stated free-energy convexity and concavity conditions, Scheme I conserves partial masses, generates nonnegative entropy production, and dissipates total energy.Source: §5.1 · pp. 16–17
Scheme I enforces quasi-incompressibility pointwise, unconditionally in one stated case or subject to a time-step condition in another; with nonnegative partial densities, the constraint yields total-density bounds.Source: §5.1 · pp. 17–18

5.2. Scheme II.

Scheme II's continuous Petrov–Galerkin discretisation generates entropy, enforces pointwise quasi-incompressibility, exactly conserves total energy, and supports arbitrary polynomial degrees at greater cost.Source: §5.2 · p. 19
In Scheme II, most time integrals can use quadrature without affecting thermodynamic stability, but exact evaluation is needed for two integrals in the energy-structure proof.In practice, sufficiently high-degree quadrature keeps the resulting energy error below that from inexact nonlinear solves.Source: §5.2 · p. 21
For Scheme II with k = 1, pointwise quasi-incompressibility holds under a time-step condition; a stated density approximation removes that restriction but may affect convergence rate.Source: §5.2 · pp. 21–24

6. Numerical simulations

The unit-square tests show spatial convergence and verify the schemes’ structure-preserving behavior; a benzene–cyclohexane application illustrates their use in a microfluidic container.

6.1.1. Spatial convergence test.

Both schemes converge at second order for partial densities, temperature, and entropy density in L∞(0, T; L2(Ω)), and at first order for temperature and chemical potentials in L2(0, T; H1(Ω)).Pressure converges at second order in L2(0, T; L2(Ω)); velocity and modified momentum density errors decay at higher, less regular rates.Source: §6.1.1 · p. 26, Table 1 · p. 27, Table 2 · p. 27

6.2. Application example.

The application uses a two-species mixture in a microfluidic container, with benzene and cyclohexane entering through separate inlets and an outlet on the opposite side.The example extends the simulation in to incorporate time dependence and thermal effects.Source: §6.2 · p. 28

6.1.2. Structure-preserving test.

In the three-component test, both schemes produce qualitatively similar density, temperature, and velocity dynamics, with later differences most noticeable in ρ3.The initial ρ1 layers and ρ2 disc are advected by the rotating flow and smoothed by diffusion.Source: §6.1.2 · p. 28, Figure 2 · p. 29, Figure 3 · p. 30
Scheme I’s total energy decreases monotonically, while Scheme II conserves it to about 10^-13; both schemes increase total entropy, conserve partial masses to about 10^-13, and satisfy the constraint to within 10^-14 in L∞(Ω).These structure-preservation checks are reported for the three-component unit-square test.Source: §6.1.2 · p. 28, Figure 4 · p. 31, Figure 5 · p. 31

6.2. Application example.

The benzene–cyclohexane microfluidic simulation gives results in good agreement between schemes; its temperature distribution follows the inlet temperatures and its benzene streamlines agree with those reported in.The solution is approximately stationary after roughly 3 h of a 6.25 h simulation.Source: Simulation results · p. 33, Figure 6 · p. 33