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Novel linear, decoupled, and energy dissipative schemes for the Navier-Stokes-Darcy model and extension to related two-phase flow

Xiaoli Li, Jie Shen, Xinhui Wang

TL;DR

This work develops linear, decoupled prediction–correction schemes for the Navier–Stokes–Darcy model that guarantee dissipation of the original energy and require only linear solves with constant coefficients at each time step. A novel relaxation-based correction using a scalar factor $\xi$ enforces unconditional energy stability and boundedness of velocity and hydraulic head, with a rigorous error analysis provided for the first-order scheme. The methodology extends to the Cahn–Hilliard–Navier–Stokes–Darcy system, yielding a first-order scheme (and EMAC variants) that preserves the original energy dissipation. Numerical experiments, including convergence tests and multiphase flow scenarios, demonstrate accuracy, stability, and robustness across a range of challenging two-phase flow problems.

Abstract

We construct efficient original-energy-dissipative schemes for the Navier-Stokes-Darcy model and related two-phase flows using a prediction-correction framework. A new relaxation technique is incorporated in the correction step to guarantee dissipation of the original energy, thereby ensuring unconditional boundedness of the numerical solutions for velocity and hydraulic head in the $l^{\infty}(L^2)$ and $l^2(H^1)$ norms. At each time step, the schemes require solving only a sequence of linear equations with constant coefficients. We rigorously prove that the schemes dissipate the original energy and, as an example, carry out a rigorous error analysis of the first-order scheme for the Navier-Stokes-Darcy model. Finally, a series of benchmark numerical experiments are conducted to demonstrate the accuracy, stability, and effectiveness of the proposed methods.

Novel linear, decoupled, and energy dissipative schemes for the Navier-Stokes-Darcy model and extension to related two-phase flow

TL;DR

This work develops linear, decoupled prediction–correction schemes for the Navier–Stokes–Darcy model that guarantee dissipation of the original energy and require only linear solves with constant coefficients at each time step. A novel relaxation-based correction using a scalar factor enforces unconditional energy stability and boundedness of velocity and hydraulic head, with a rigorous error analysis provided for the first-order scheme. The methodology extends to the Cahn–Hilliard–Navier–Stokes–Darcy system, yielding a first-order scheme (and EMAC variants) that preserves the original energy dissipation. Numerical experiments, including convergence tests and multiphase flow scenarios, demonstrate accuracy, stability, and robustness across a range of challenging two-phase flow problems.

Abstract

We construct efficient original-energy-dissipative schemes for the Navier-Stokes-Darcy model and related two-phase flows using a prediction-correction framework. A new relaxation technique is incorporated in the correction step to guarantee dissipation of the original energy, thereby ensuring unconditional boundedness of the numerical solutions for velocity and hydraulic head in the and norms. At each time step, the schemes require solving only a sequence of linear equations with constant coefficients. We rigorously prove that the schemes dissipate the original energy and, as an example, carry out a rigorous error analysis of the first-order scheme for the Navier-Stokes-Darcy model. Finally, a series of benchmark numerical experiments are conducted to demonstrate the accuracy, stability, and effectiveness of the proposed methods.
Paper Structure (20 sections, 8 theorems, 92 equations, 9 figures)

This paper contains 20 sections, 8 theorems, 92 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The Navier--Stokes--Darcy model
  3. Model definition and concepts
  4. Energy dissipation law
  5. Energy stable schemes
  6. Scheme I: a first-order scheme
  7. Scheme II: a second-order scheme
  8. Error analysis
  9. A rough error estimate
  10. A refined error estimate
  11. Extension to the Cahn--Hilliard--Navier--Stokes--Darcy model
  12. Numerical experiments
  13. The EMAC formulation
  14. Example 1: Convergence tests A
  15. Example 2: Convergence tests B
  16. ...and 5 more sections

Key Result

Lemma 2.1

\newlabela=0 Let $\mathbf{u}\in H_f$ and $\nabla\cdot \mathbf{u}=0$, then, we have wang2024class

Figures (9)

  • Figure 6.1: Comparison of numerical errors and convergence rates with $T=0.5$ (Example 1). Left: Scheme I. Right: Scheme II.
  • Figure 6.2: Comparison of numerical errors and convergence rates with $T=0.5$ (Example 2). Left: Scheme I. Right: Scheme II.
  • Figure 6.3: Comparison of velocity field distributions driven by initial velocity in different porous-media configurations.
  • Figure 6.4: Left: The velocity field distributions with random hydraulic conductivity tensor; Right: The comparison of $\xi$ with different settings.
  • Figure 6.5: Phase-field evolution from random initial conditions at T=0.5, 1, 2, 4, 10.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • ...and 3 more