Analysis of second order and unconditionally stable BDF2‐AB2 method for the Navier‐Stokes equations with nonlinear time relaxation

Abstract

In this study, we first consider a second order time stepping finite element BDF2‐AB2 method for the Navier‐Stokes equations (NSE). We prove that the method is unconditionally stable and O ( Δ t 2 ) accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term “ κ | u − u ¯ | ( u − u ¯ ) ” to the Navier‐Stokes Equations with the algorithm depends on BDF2‐AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2‐AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2‐AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2‐AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.

Higher-order time-stepping schemes for fluid-structure interaction problems

“…When applied to ODEs, it is well-known that BDF2 is convergent of order 2 and A-stable, (see for more detail [14,Chapter 7]). BDF methods have been successfully applied to the Navier-Stokes equations [20], to nonlinear structural mechanics [15], to electro-magnetic problems [22], and to Stokes-Darcy flow [11].…”

“…We mimic the proof of [3,Prop. 3] and use some useful tricks presented in [20]. We test (26a) with u n+1…”

Higher-order time-stepping schemes for fluid-structure interaction problems

“…Remark. We can introduce an alternative to the third equation in (16) by splitting it into a system of two equations of first order in time, and by introducing a new unknown Ẋ(t) = ∂X(t) ∂t :…”

“…By choosing properly the test functions in (16), one can obtain the following energy estimate [5] : Proposition 2.1. Let us assume that δρ ≥ 0, that the potential energy density W is a C 1 convex function over the set of second order tensors, and that for almost every t ∈ [0, T ] the solution of Problem 1 is such that X(t) ∈ (W In the above proposition • D stands for the norm in L 2 (D).…”

Solutions and Stability Analysis of Backward-Euler Method for Simplified Magnetohydrodynamics With Nonlinear Time Relaxation

“…In [14], Pakzad examines time averaged energy dissipation rate for the Time Relaxation Model for 3d turbulence with periodic boundary conditions. The studies about the time relaxation regularization can be found in the literature, such as [15,16,17]. The effects of the linear and nonlinear time relaxation terms to SMHD equations with BE finite element method is presented in [18].…”