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

Rong, Yao; Fiordilino, Joseph A.

doi:10.1007/s10915-020-01165-5

Cited by 20 publications

(7 citation statements)

References 33 publications

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“… 2014 ; Neilan and Zytoon 2020 ; Qin et al. 2020 ; Rong and Fiordilino 2020 ). Thanks to this stabilization term, we have the stability for the Darcy–Brinkman equation (see Xie et al.…”

Section: Numerical Simulationmentioning

confidence: 99%

A Mathematical Description of the Flow in a Spherical Lymph Node

2022

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The motion of the lymph has a very important role in the immune system, and it is influenced by the porosity of the lymph nodes: more than 90% takes the peripheral path without entering the lymphoid compartment. In this paper, we construct a mathematical model of a lymph node assumed to have a spherical geometry, where the subcapsular sinus is a thin spherical shell near the external wall of the lymph node and the core is a porous material describing the lymphoid compartment. For the mathematical formulation, we assume incompressibility and we use Stokes together with Darcy–Brinkman equation for the flow of the lymph. Thanks to the hypothesis of axisymmetric flow with respect to the azimuthal angle and the use of the stream function approach, we find an explicit solution for the fully developed pulsatile flow in terms of Gegenbauer polynomials. A selected set of plots is provided to show the trend of motion in the case of physiological parameters. Then, a finite element simulation is performed and it is compared with the explicit solution.

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“… 2014 ; Neilan and Zytoon 2020 ; Qin et al. 2020 ; Rong and Fiordilino 2020 ). Thanks to this stabilization term, we have the stability for the Darcy–Brinkman equation (see Xie et al.…”

Section: Numerical Simulationmentioning

confidence: 99%

A Mathematical Description of the Flow in a Spherical Lymph Node

2022

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“…This section makes the method and result precise and proves stability for α ≥ 0.5γ and control of ∇ • u for α > 0.5γ for the modular sparse graddiv algorithm. This work builds on Guermond and Minev [10], the work on modular grad-div in [8] and Rong and Fiordilino [20] and the numerical tests of a related method in Demir and Kaya [7]. We suppress the traditional sub-or super-scripts "h" in finite element formulations.…”

Section: Analysis Of Modular Sparse Grad-divmentioning

confidence: 99%

Stability in 3d of a sparse grad-div approximation of the Navier-Stokes equations

Layton¹

2021

Preprint

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Inclusion of a term −γ∇∇ • u, forcing ∇ • u to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let G * = −diag(∂ 2x , ∂ 2 y , ∂ 2 z ) denote the diagonal of G = −∇∇•, and α ≥ 0 an adjustable parameter. The 2-step method considered is 1 :We prove its unconditional, nonlinear, long time stability in 3d for α ≥ 0.5γ. The analysis also establishes that the method controls the persistent size of ∇ • u in general and controls the transients in ∇ • u for a cold start when α > 0.5γ. Consistent numerical tests are presented.

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“…In this validation experiment, we consider the famous Laminar flow past a cylinder [50], which is also called the Schäfer-Turek benchmark. It is popular with the literature about the time-dependent Navier-Stokes equations [14,27,39,49]. Our purpose of solving the stationary equations is to investigate the appearance and evolution of the symmetric vortices behind the cylinder as the Reynolds number increases.…”

Section: Example 3: Robustness For Irrotational Body Forcesmentioning

confidence: 99%

Analysis and computation of a pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations by using high-order finite elements

Yang,

2021

Preprint

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In this work, we develop a high-order pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H(div)-conforming velocity reconstruction operator. In order to match the rotation form and error analysis, a novel skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proven to achieve the pressure-independent velocity errors. Optimal convergence rates of H 1 , L 2 -error for the velocity and L 2 -error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and the remarkable performance of the proposed method.

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

Cited by 20 publications

References 33 publications

A Mathematical Description of the Flow in a Spherical Lymph Node

A Mathematical Description of the Flow in a Spherical Lymph Node

Stability in 3d of a sparse grad-div approximation of the Navier-Stokes equations

Analysis and computation of a pressure-robust method for the rotation form of the stationary incompressible Navier-Stokes equations by using high-order finite elements

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