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Artificial compressibility method for the incompressible Navier-Stokes equations with variable density

Cappanera Loic, Giordano Salvatore

TL;DR

This work develops an artificial compressibility framework for incompressible Navier–Stokes equations with variable density and viscosity, formulating the momentum and pressure as coupled primary unknowns to avoid Poisson solves and enable time‑independent stiffness suitable for spectral methods. Stability and temporal convergence are established for a semi‑implicit scheme under a minimum‑maximum density principle, with velocity error shown to be of order $O( au^{1/2})$ in theory (and often $O( au)$ in practice), and an explicit variant demonstrated for spectral codes that retains first‑order accuracy under CFL. Numerical experiments using level‑set density tracking validate robustness for both linear and nonlinear $ u( ho)$, include smooth and nonsmooth interfaces, and compare favorably to momentum‑based projection methods, especially in complex multiphase/gravitational wave scenarios. The paper also discusses normalization of the artificial compressibility parameter and outlines future directions toward higher‑order and unconditionally stable formulations (e.g., SAV approaches).

Abstract

We introduce a novel artificial compressibility technique to approximate the incompressible Navier-Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed scheme used the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a scheme with time independent stiffness matrices that is suitable for pseudo-spectral methods. The stability and temporal convergence of the semi-implicit version of the scheme is established under the hypothesis that the density is approximated with a method that conserves the minimum-maximum principle. Numerical illustrations confirm that both the semi-implicit and explicit scheme are stable and converge with order one under classic CFL condition. Moreover, the proposed scheme is shown to perform better than a momentum based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi-fluids in a cylinder.

Artificial compressibility method for the incompressible Navier-Stokes equations with variable density

TL;DR

This work develops an artificial compressibility framework for incompressible Navier–Stokes equations with variable density and viscosity, formulating the momentum and pressure as coupled primary unknowns to avoid Poisson solves and enable time‑independent stiffness suitable for spectral methods. Stability and temporal convergence are established for a semi‑implicit scheme under a minimum‑maximum density principle, with velocity error shown to be of order in theory (and often in practice), and an explicit variant demonstrated for spectral codes that retains first‑order accuracy under CFL. Numerical experiments using level‑set density tracking validate robustness for both linear and nonlinear , include smooth and nonsmooth interfaces, and compare favorably to momentum‑based projection methods, especially in complex multiphase/gravitational wave scenarios. The paper also discusses normalization of the artificial compressibility parameter and outlines future directions toward higher‑order and unconditionally stable formulations (e.g., SAV approaches).

Abstract

We introduce a novel artificial compressibility technique to approximate the incompressible Navier-Stokes equations with variable fluid properties such as density and dynamical viscosity. The proposed scheme used the couple pressure and momentum, equal to the density times the velocity, as primary unknowns. It also involves an adequate treatment of the diffusive operator such that treating the nonlinear convective term explicitly leads to a scheme with time independent stiffness matrices that is suitable for pseudo-spectral methods. The stability and temporal convergence of the semi-implicit version of the scheme is established under the hypothesis that the density is approximated with a method that conserves the minimum-maximum principle. Numerical illustrations confirm that both the semi-implicit and explicit scheme are stable and converge with order one under classic CFL condition. Moreover, the proposed scheme is shown to perform better than a momentum based pressure projection method, previously introduced by one of the authors, on setups involving gravitational waves and immiscible multi-fluids in a cylinder.

Paper Structure

This paper contains 26 sections, 6 theorems, 135 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Problem description and artificial compressibility method
  3. The incompressible Navier-Stokes equations with variable density and viscosity
  4. Notations and preliminaries
  5. Artificial compressibility method based on momentum approximation
  6. Stability of a semi-implicit algorithm
  7. Approximation of Density and maximum principle hypothesis.
  8. Time marching algorithm
  9. Stability of the algorithm
  10. Discussion on normalization of artificial compressibility parameter
  11. Time error analysis
  12. Notations and hypothesis
  13. Preliminaries results
  14. Main results
  15. Numerical Illustrations
  16. ...and 11 more sections

Key Result

theorem 1

Let $\gamma_1\in(0,\frac{1}{\sqrt{2}})$, $\gamma_2\in(0,\frac{1}{\sqrt{2} c_3})$ and $(\gamma_3,\gamma_4)\in(0,1)^2$ be constants. Let $\bar{\nu}$ and $\bar{\lambda}$ satisfy eq:def_nu_rho_bar. Assume that the approximation of the density $\rho^\tau$ is given by an algorithm of the form eq:approxima Also assume that the sequence $\rho^\tau$ satisfies: Then the sequences $(\mathbf{m}^\tau, p^\tau,

Theorems & Definitions (16)

  • theorem 1
  • remark 1
  • proof
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • theorem 2
  • ...and 6 more