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A class of new linear, efficient and high-order implicit-explicit methods for the coupled free flow-porous media system based on nonlinear Lions interface condition

Xinhui Wang, Xu Guo, Xiaoli Li

TL;DR

The paper addresses the challenge of simulating the coupled free-flow and porous-media system governed by unsteady NS–Darcy equations with the Lions interface condition. It introduces linear, high-order IMEX schemes based on the scalar auxiliary variable (SAV) approach in time and finite elements in space, yielding decoupled, linear solves for the Navier–Stokes and Darcy subsystems while preserving unconditional stability. A rigorous error analysis for the first-order scheme is provided, leveraging Ritz projections and Gronwall-type arguments under regularity assumptions. Numerical tests, including convergence studies and representative flow scenarios, confirm the theoretical predictions and demonstrate the methods' efficiency and accuracy for practical coupled-flow problems.

Abstract

In this paper, we construct and analyze new first- and second-order implicit-explicit (IMEX) schemes for the unsteady Navier-Stokes-Darcy model to describe the coupled free flow-porous media system, which is based on the scalar auxiliary variable (SAV) approach in time and finite element method in space. The constructed schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and can decouple the Navier-Stokes and Darcy systems. The unconditional stability of both the first- and second-order IMEX schemes can be derived for the coupled system equipped with the Lions interface condition, where the key point is that we should construct a new trilinear form to balance the fully explicit discretizations of the nonlinear terms in the complex system. We can also establish rigorous error estimates for the velocity and hydraulic head of the first-order scheme without any time step restriction. Numerical examples are presented to validate the proposed schemes.

A class of new linear, efficient and high-order implicit-explicit methods for the coupled free flow-porous media system based on nonlinear Lions interface condition

TL;DR

The paper addresses the challenge of simulating the coupled free-flow and porous-media system governed by unsteady NS–Darcy equations with the Lions interface condition. It introduces linear, high-order IMEX schemes based on the scalar auxiliary variable (SAV) approach in time and finite elements in space, yielding decoupled, linear solves for the Navier–Stokes and Darcy subsystems while preserving unconditional stability. A rigorous error analysis for the first-order scheme is provided, leveraging Ritz projections and Gronwall-type arguments under regularity assumptions. Numerical tests, including convergence studies and representative flow scenarios, confirm the theoretical predictions and demonstrate the methods' efficiency and accuracy for practical coupled-flow problems.

Abstract

In this paper, we construct and analyze new first- and second-order implicit-explicit (IMEX) schemes for the unsteady Navier-Stokes-Darcy model to describe the coupled free flow-porous media system, which is based on the scalar auxiliary variable (SAV) approach in time and finite element method in space. The constructed schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and can decouple the Navier-Stokes and Darcy systems. The unconditional stability of both the first- and second-order IMEX schemes can be derived for the coupled system equipped with the Lions interface condition, where the key point is that we should construct a new trilinear form to balance the fully explicit discretizations of the nonlinear terms in the complex system. We can also establish rigorous error estimates for the velocity and hydraulic head of the first-order scheme without any time step restriction. Numerical examples are presented to validate the proposed schemes.
Paper Structure (12 sections, 5 theorems, 84 equations, 9 figures, 2 tables)

This paper contains 12 sections, 5 theorems, 84 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The unsteady Navier-Stokes-Darcy model
  3. Numerical schemes
  4. First-order scheme
  5. Second-order scheme
  6. Error Analysis
  7. Numerical Tests
  8. Convergence test
  9. Simulation of fluid flow in a Y-shape domain
  10. Filtration processes
  11. Cavity flow
  12. Conclusion

Key Result

Lemma 2.1

\newlabellemma 1 For the trilinear term $a_N(u,v,w)$, the following inequalities hold shen1992onConnors2012fluid: and when take $d=2$, where $C$ is a positive constant which is dependent on $\Omega_f$.

Figures (9)

  • Figure 2.1: Coupled domains of free flow and porous media flow
  • Figure 5.1: Schematic diagram of the Y-shape domain and mesh partition.
  • Figure 5.2: Evolutions of velocity and pressure for Algorithm 1 ($\omega_1=0.5,~\omega_2=1,~k=1$).
  • Figure 5.3: Evolutions of velocity and pressure for Algorithm 1 ($\omega_1=0.5,~\omega_2=1,~k=10^{-2}$).
  • Figure 5.4: Evolutions of velocity and pressure for Algorithm 1 ($\omega_1=0.5,~\omega_2=1,~k=10^{-4}$).
  • ...and 4 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof