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

TL;DR

This paper develops and analyzes an implicit-explicit Crank-Nicolson scheme for high Reynolds number Oseen's equations, ensuring stability and accuracy through stabilization techniques and providing numerical comparisons.

Contribution

The paper introduces a stabilized IMEX Crank-Nicolson scheme for high Reynolds number Oseen's equations, with proven stability and error estimates, extending previous work on IMEX methods.

Findings

Stability of the scheme under Courant conditions.

Error estimates of order O(h^{k+1/2} + τ^2).

Numerical comparisons of different methods on Navier-Stokes equations.

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. Guzm\`an, {\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 $O(h^{k+\frac12} + \tau^2)$. Here $h$ is the mesh parameter, $k$ the polynomial order and $\tau$ 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.