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Long-time stable SAV-BDF2 numerical schemes for the forced Navier-Stokes equations

Daozhi Han, Xiaoming Wang

Abstract

We propose a novel second-order accurate, long-time unconditionally stable time-marching scheme for the forced Navier-Stokes equations. A new Forced Scalar Auxiliary Variable approach (FSAV) is introduced to preserve the underlying dissipative structure of the forced system that yields a uniform-in-time estimate of the numerical solution. In addition, the numerical scheme is autonomous if the underlying model is, laying the foundation for studying long-time dynamics of the numerical solution via dynamical system approach. As an example we apply the new algorithm to the two-dimensional incompressible Navier-Stokes equations. In the case with no-penetration and free-slip boundary condition on a simply connected domain, we are also able to derive a uniform-in-time estimate of the vorticity in $H^1$ norm in addition to the $L^2$ norm guaranteed by the general framework. Numerical results demonstrate superior performance of the new algorithm in terms of accuracy, efficiency, stability and robustness.

Long-time stable SAV-BDF2 numerical schemes for the forced Navier-Stokes equations

Abstract

We propose a novel second-order accurate, long-time unconditionally stable time-marching scheme for the forced Navier-Stokes equations. A new Forced Scalar Auxiliary Variable approach (FSAV) is introduced to preserve the underlying dissipative structure of the forced system that yields a uniform-in-time estimate of the numerical solution. In addition, the numerical scheme is autonomous if the underlying model is, laying the foundation for studying long-time dynamics of the numerical solution via dynamical system approach. As an example we apply the new algorithm to the two-dimensional incompressible Navier-Stokes equations. In the case with no-penetration and free-slip boundary condition on a simply connected domain, we are also able to derive a uniform-in-time estimate of the vorticity in norm in addition to the norm guaranteed by the general framework. Numerical results demonstrate superior performance of the new algorithm in terms of accuracy, efficiency, stability and robustness.
Paper Structure (11 sections, 4 theorems, 69 equations, 7 figures, 3 tables)

This paper contains 11 sections, 4 theorems, 69 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The numerical scheme
  3. The general scheme
  4. The scheme for the 2D Navier-Stokes equations
  5. Error analysis
  6. Numerical experiments
  7. Accuracy
  8. The Kolmogorov flow
  9. Long-time stability
  10. Bursting
  11. Conclusion

Key Result

Theorem 2.1

Suppose $\bm{u}_0 \in \mathbf{H}^1_0(\Omega), \mathbf{F} \in \mathbf{L}^\infty(0,\infty; \mathbf{L}^2(\Omega))$ and the scheme DDSe-NS--DSAV-NS is initiated by the first order counterpart. Then for any $k >0$ the scheme DDSe-NS --DSAV-NS is long-time stable in the sense that $||\bm{u}^n||+|q^n|\leq

Figures (7)

  • Figure 1: The $L^2$ norm and $H^1$ norm of the vorticity as a function of time by the SAV-BDF2 scheme with 256 Fourier modes and $k=0.01, 0.005, 0.0025$ respectively.
  • Figure 2: The real and imaginary part of the Fourier coefficient of mode $e^{iy}$ as a function of time. $Re=25.7715, k=0.001$ with 256 Fourier modes.
  • Figure 3: The largest magnitude and the $L^2$ norm of gradient of vorticity as a function of time. $Re=25.7715, k=0.001$ with 256 Fourier modes.
  • Figure 4: Time interval between bursts
  • Figure 5: Power spectrum density plot
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof