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Implicit-explicit Crank-Nicolson scheme for Oseen's equation at high Reynolds number

Erik Burman, Deepika Garg, Johnny Guzman

TL;DR

This work develops a stabilized implicit–explicit Crank–Nicolson (IMEX CN) scheme for the transient Oseen equations at high Reynolds number, treating viscous and pressure–velocity coupling implicitly while handling convection explicitly through extrapolation. By employing a symmetric gradient-jump (CIP-like) stabilization with equal-order finite elements, the authors ensure inf–sup compatibility and robust performance as the mesh Reynolds number grows, under appropriate Courant conditions. They prove stability and derive a priori error estimates of order $O(h^{k+\frac{1}{2}}+\tau^2)$, and they develop splitting variants derived from the IMEX approach that yield decoupled Poisson-pressure projections compatible with viscous and inviscid limits. Numerical experiments including Taylor–Green vortex, high-Re mixing layers, and cylinder flow validate the theory, showing competitive accuracy and robustness, while highlighting the splitting scheme’s dissipative effects in transitional regimes.

Abstract

In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzmàn, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order $O(h^{k+\frac12} + τ^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $τ$ the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations.

Implicit-explicit Crank-Nicolson scheme for Oseen's equation at high Reynolds number

TL;DR

This work develops a stabilized implicit–explicit Crank–Nicolson (IMEX CN) scheme for the transient Oseen equations at high Reynolds number, treating viscous and pressure–velocity coupling implicitly while handling convection explicitly through extrapolation. By employing a symmetric gradient-jump (CIP-like) stabilization with equal-order finite elements, the authors ensure inf–sup compatibility and robust performance as the mesh Reynolds number grows, under appropriate Courant conditions. They prove stability and derive a priori error estimates of order , and they develop splitting variants derived from the IMEX approach that yield decoupled Poisson-pressure projections compatible with viscous and inviscid limits. Numerical experiments including Taylor–Green vortex, high-Re mixing layers, and cylinder flow validate the theory, showing competitive accuracy and robustness, while highlighting the splitting scheme’s dissipative effects in transitional regimes.

Abstract

In this paper we continue the work on implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations that we started in \cite{BGG23} (E. Burman, D. Garg, J. Guzmàn, {\emph{Implicit-explicit time discretization for Oseen's equation at high Reynolds number with application to fractional step methods}}, SIAM J. Numer. Anal., 61, 2859--2886, 2023). The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein we focus on the implicit-explicit Crank-Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions we prove stability of the implicit-explicit Crank-Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order . Here is the mesh parameter, the polynomial order and the time step. Finally we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier-Stokes' equations.
Paper Structure (16 sections, 17 theorems, 156 equations, 8 figures, 2 algorithms)

This paper contains 16 sections, 17 theorems, 156 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Weak formulation and space semi-discretization
  4. Fully discrete methods
  5. Technical Results
  6. Stability and error analysis, Crank-Nicolson method with extrapolation
  7. Stability Crank-Nicolson, general case
  8. Improved stability for piece wise affine elements and CIP stabilization, Crank-Nicolson
  9. A priori error estimate for the velocity approximation
  10. Error estimates for the pressure approximation
  11. Splitting methods derived from the IMEX method
  12. Numerical examples
  13. Convergence Study: Taylor--Green vortex
  14. Navier--Stokes' mixing layer at $\Re=10^4$
  15. Splitting scheme with known solution $Re<1$
  16. ...and 1 more sections

Key Result

Lemma 2.1

\newlabelconsistency ( modified Galerkin orthogonality). Assume that $(u, p)$, the solution of (eq:convdiff), belongs to the space $[H^{\frac{3}{2}+\epsilon}(\Omega)]^{d+1}$, with $\epsilon > 0$, and let $(u_h, p_h) \in V_h \times Q_h$ be the solution of (eq:scheme_semi). Then for all $(v_h, q_h) \in V_h \times Q_h$.

Figures (8)

  • Figure 6.1: Convergence plot of the Crank-Nicolson discretization applied to the problem defined by the exact solution (\ref{['exact_sol']}). Dashed (resp. Dotted) lines represented the CN- IMEX (resp. CN-Split) scheme. Triangle markers describe the $L^2$-error of the velocity, and squares markers represent pressure. Solid lines mark depicts the slopes. left: $P_1$ approximation of velocity and pressure; right: $P_2$ approximation of velocity and pressure.
  • Figure 6.2: Computational configuration for the Kelvin--Helmholtz shear layer instability
  • Figure 6.3: Time evolution of (left to right) the kinetic energy, artificial dissipation and physical dissipation in computational mesh: $80 \times 80$; piecewise quadratic approximation.
  • Figure 6.4: Comparison of CN-imex (a), CN-split (b) ; CN-monolith (c); time levels, from left to right, $t=80, 120, 140$; computational mesh: $80 \times 80$; piecewise quadratic approximation.
  • Figure 6.5: Convergence plot of the Crank-Nicolson splitting scheme in the viscous flow applied to the problem defined by the exact solution (\ref{['exact_sol_ex2']}). Absolute-error. Left $P_1$ Approximations. Right $P_2$ Approximations.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • ...and 18 more