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A structure-preserving numerical method for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems

Aaron Brunk, Ansgar Jüngel, Maria Lukáčová-Medvid'ová

TL;DR

The paper develops a conforming finite element method with mixed explicit-implicit time discretization for the quasi-incompressible Navier–Stokes–Maxwell–Stefan system, ensuring mass preservation, pointwise quasi-incompressibility, and a discrete energy inequality. It provides a rigorous variational formulation, derives discrete identities and a priori bounds, and demonstrates convergence and structure-preserving behavior through two- and three-component 2D experiments, including a relative energy analysis. The work advances numerical analysis and simulation capability for multicomponent, cross-diffusion flows by delivering a robust, energy-stable discretization that respects the coupled physics. This framework lays groundwork for error estimates and higher-order structure-preserving schemes, with potential applications to complex fluids and electrolyte transport problems.

Abstract

A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier-Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell-Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties.

A structure-preserving numerical method for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems

TL;DR

The paper develops a conforming finite element method with mixed explicit-implicit time discretization for the quasi-incompressible Navier–Stokes–Maxwell–Stefan system, ensuring mass preservation, pointwise quasi-incompressibility, and a discrete energy inequality. It provides a rigorous variational formulation, derives discrete identities and a priori bounds, and demonstrates convergence and structure-preserving behavior through two- and three-component 2D experiments, including a relative energy analysis. The work advances numerical analysis and simulation capability for multicomponent, cross-diffusion flows by delivering a robust, energy-stable discretization that respects the coupled physics. This framework lays groundwork for error estimates and higher-order structure-preserving schemes, with potential applications to complex fluids and electrolyte transport problems.

Abstract

A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier-Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell-Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties.

Paper Structure

This paper contains 14 sections, 6 theorems, 55 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model equations
  3. Main results
  4. State of the art
  5. Various formulations
  6. Numerical analysis
  7. Numerical scheme
  8. Properties of the numerical scheme
  9. Remarks
  10. Numerical experiments
  11. Convergence tests
  12. Three-component experiment I
  13. Three-component experiment II
  14. Conclusion and outlook

Key Result

lemma thmcounterlemma

Equations 1.mass--1.flux, 1.chem--1.incom can be written as 1.mass2--1.divMu for smooth solutions. If $V_i=V>0$ for $i=1,\ldots,N$, equations 1.mom2 and 1.divMu reduce to 1.momim.

Figures (5)

  • Figure 1: Snapshots of the partial mass densities $\rho_1$ (upper row), $\rho_2$ (middle row), and $\rho_3$ (lower row) at times $t=0.001,0.02,0.04,0.06, 0.1,0.5$ for the experiment in Section \ref{['subsec:experiment']}.
  • Figure 2: Snapshots of the total mass density $\rho$ (upper row), velocity magnitude $|\bm{u}|^2$, and pressure $p$ at times $t=0.001,0.02,0.04,0.06,0.1,0.5$ for the experiment in Section \ref{['subsec:experiment']}.
  • Figure 3: Relative energy versus time for the experiment in Section \ref{['subsec:experiment']}.
  • Figure 4: Snapshots of the partial mass densities $\rho_{1}$ (first row), $\rho_2$ (second row), $\rho_3$ (third row) and the total mass density $\rho$ (last row) at times $t=0.1,0.6,1,2$. for the experiment in Section \ref{['subsec:experiment2']}.
  • Figure 5: Relative energy versus time for the experiment in Section \ref{['subsec:experiment2']}.

Theorems & Definitions (14)

  • lemma thmcounterlemma: Reformulation
  • proof
  • lemma thmcounterlemma: Positivity of the mass densities
  • proof
  • lemma thmcounterlemma: Energy identity
  • proof
  • lemma thmcounterlemma: Variational formulation
  • proof
  • theorem 1: Discrete identities
  • proof
  • ...and 4 more