Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unconditional stability and convergence analysis of novel regularization schemes for the Navier-Stokes equations

Zhaoyang Wang, Ping Lin

Abstract

In this paper, we construct novel first- and second-order decoupled schemes for the Navier-Stokes equations based on the penalty method and the sequential regularization method (SRM), respectively. These schemes do not require the boundary condition on the pressure and thus preserve the original velocity boundary conditions. By using the idea of the scalar auxiliary variable (SAV), the nonlinear terms of these schemes are treated explicitly, which improves computational efficiency while maintaining stability. It is important to note that we carefully reformulated the Navier-Stokes system to ensure convergence of the proposed scheme without any restriction on the time step. For the Penalty-SAV (P-SAV) schemes, at each time step it is only necessary to solve elliptic equations with constant coefficients. We prove the high-order stability (high-order regularities of the solution) of the schemes, and establish an unconditional (without time step constraints) global optimal error estimate in two dimensions as well as a local error estimate in three dimensions for the first-order scheme. Furthermore, to more accurately approximate the incompressibility constraint without introducing extra stiffness into the system, the sequential regularization-SAV (SR-SAV) schemes are developed, and their error estimates are provided. In addition, we compare our proposed scheme with the classic linearized projection scheme to demonstrate its accuracy and efficiency.

Unconditional stability and convergence analysis of novel regularization schemes for the Navier-Stokes equations

Abstract

In this paper, we construct novel first- and second-order decoupled schemes for the Navier-Stokes equations based on the penalty method and the sequential regularization method (SRM), respectively. These schemes do not require the boundary condition on the pressure and thus preserve the original velocity boundary conditions. By using the idea of the scalar auxiliary variable (SAV), the nonlinear terms of these schemes are treated explicitly, which improves computational efficiency while maintaining stability. It is important to note that we carefully reformulated the Navier-Stokes system to ensure convergence of the proposed scheme without any restriction on the time step. For the Penalty-SAV (P-SAV) schemes, at each time step it is only necessary to solve elliptic equations with constant coefficients. We prove the high-order stability (high-order regularities of the solution) of the schemes, and establish an unconditional (without time step constraints) global optimal error estimate in two dimensions as well as a local error estimate in three dimensions for the first-order scheme. Furthermore, to more accurately approximate the incompressibility constraint without introducing extra stiffness into the system, the sequential regularization-SAV (SR-SAV) schemes are developed, and their error estimates are provided. In addition, we compare our proposed scheme with the classic linearized projection scheme to demonstrate its accuracy and efficiency.

Paper Structure

This paper contains 13 sections, 10 theorems, 103 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The Penalty-SAV system and its convergence
  3. System reconstruction and preliminaries
  4. Error estimate of the P-SAV formulation
  5. The numerical schemes and their stability
  6. Numerical schemes
  7. Strong stability results
  8. Error analysis
  9. SAV with a sequential regularization method (SR-SAV method)
  10. Error estimate
  11. Numerical schemes
  12. Numerical experiments
  13. Conclusions and remarks

Key Result

Lemma 2.1

Let $G:(0,\infty)\rightarrow(0,\infty)$ be continuous and increasing, and let $T^*$ satisfy that $0<T^*<\int_{M}^{\infty} dz/G(z)$ for some $M>0$. Suppose that quantities $z_n, w_n\geq 0$ satisfy with $n^{*}\Delta t\leq T^*$. Then we have $M+\sum\limits_{i=0}^{n-1}\Delta t G(z_i)\leq C_{*}$, where $C_{*}$ is independent of $\Delta t$.

Figures (2)

  • Figure 1: Evolution of energy for the second-order P-SAV scheme with $\Delta t=1/4$ and $\Delta t=1/32$.
  • Figure 2: Snapshots of the velocity field of flow passing a cylinder.

Theorems & Definitions (22)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • ...and 12 more