An efficient and modular grad–div stabilization

Abstract

This paper presents two modular grad-div algorithms for calculating solutions to the Navier-Stokes equations (NSE). These algorithms add to an NSE code a minimally intrusive module that implements grad-div stabilization. The algorithms do not suffer from either breakdown (locking) or debilitating slow down for large values of grad-div parameters. Stability and optimal-order convergence of the methods are proven. Numerical tests confirm the theory and illustrate the benefits of these algorithms over a fully coupled grad-div stabilization.

Numerical Analysis of a BDF2 Modular Grad–Div Stabilization Method for the Navier–Stokes Equations

“…Observing the curves of different γ and β, it's interesting to find that the value of β determines the minimum divergence error that can be reached in the beginning and the value of γ determines the long-time divergence error. This is consistent with [5]. In Figure 5.2, we see that results for Step 2 of BDF2-mgd are consistent with Standard Stablilzed; both reduce divergence error, especially around the step.…”

“…The pressure difference between the front and back of the cylinder (∆p(t) = p(0.15, 0.2, t) − p(0.25, 0.2, t)) and both the L 2 (0, T ; L 2 (Ω)) and L ∞ (0, T ; L 2 (Ω)) norms of the velocity divergence are also tabulated in Table 5.4. Furthermore, Figure 5.3 shows velocity speed and vectors for BDF2-mgd at times t = 4, 6, 7, 8, which are consistent with that in [2,5,13,21].…”

“…Proof. The proof follows by similar arguments as in Theorem 5 of [5]. Next, we analyze the stability of BDF2-mgd.…”

“…. The channel considered here is [0, 40] × [0, 10] with a 1 × 1 step on the bottom for x ∈[5,6]. A flow with ν = 1/600 passes though this channel from left to right.…”

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model

“…However, this method leads to a singular matrix stemming from grad-div term, and the larger stabilized parameter will cause solver breakdown. As a alternative method for grad-div stabilization, the modular graddiv stabilization method is introduced in [3]. The modular grad-div stabilization method for the Stokes/Darcy model is proposed in [13].…”