On stability and convergence of projection methods based on pressure Poisson equation

Guermond, Jean‐Luc; Quartapelle, L.

doi:10.1002/(sici)1097-0363(19980515)26:9<1039::aid-fld675>3.0.co;2-u

Cited by 169 publications

(114 citation statements)

References 16 publications

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“…It is worth noting that the "mini" element satisfies the LBB (inf-sup) condition, while the equal-order ( 1 / 1 ) approximation does not. However, in combination with the operator-splitting (8)-(9) (which is, in fact, a continuous version of the well-known Chorin projection scheme), the 1 / 1 approximation is stable, if the time step ∆t ≥ Ch 2 (C is a constant), see, e.g., Guermond & Quartapelle (1998). This fact enables us to use the most economical but sufficiently accurate 1 / 1 approximation for the velocity and pressure.…”

Section: Discretization Of the Navier-stokes Equationsmentioning

confidence: 99%

Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces

Smolianski

2005

Int. J. Numer. Meth. Fluids

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The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modeling of twofluid interfacial flows, having in mind possible interface topology changes (like merger or break-up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator-splitting for temporal discretization and the level-set method for interface representation. We show that the finite element implementation of the level-set approach brings some additional benefits as compared to the standard, finite difference level-set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second-order accuracy of the interface normal, curvature and mass conservation. The operator-splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal-order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh-Taylor instability are presented to validate the computational method.

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Section: Discretization Of the Navier-stokes Equationsmentioning

confidence: 99%

Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces

Smolianski

2005

Int. J. Numer. Meth. Fluids

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“…We illustrate the convergence rate of (2.8) by considering a test problem for the NSE from [8], which has an analytical solution. The problem preserves spatial patterns of the Green-Taylor solution [1,9] but the vortices do not decay as t → ∞.…”

Section: Convergence Testmentioning

confidence: 99%

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

Gunzburger

Jiang

Wang

2017

Computational Methods in Applied Mathematics

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Abstract. We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple righthand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.Key words. Navier-Stokes equations, parameterized flow, ensemble method MSC (2010). 65M60, 76D051. Introduction. Many computational fluid dynamics applications require multiple simulations of a flow under different input conditions. For example, the ensemble Kalman filter approach used in data assimilation first simulates a forward model a large number of times by perturbing either the initial condition data, boundary condition data, or uncertain parameters, then corrects the model based on the model forecasts and observational data. A second example is the construction of low-dimensional surrogates for solutions of partial differential equation (PDE) such as sparse-grid interpolants or proper orthogonal decomposition approximations for which one has to first obtain expensive approximations of solutions corresponding to several parameter samples. Another example is sensitivity analyses of solutions for which one often has to determine approximate solutions for a number of perturbed inputs such as the values of certain physical parameters. In this paper, we consider such applications and develop a second-order time-stepping scheme for efficiently simulating an ensemble of flows. In particular, we consider the setting in which one wishes to determine the PDE solutions for several different values of the physical parameters and with several different choices of initial condition and boundary condition data and forcing functions appearing in the PDE model.The ensemble algorithm we use was first developed in [11] to find a set of J solutions of the Navier-Stokes equations (NSE) subject to different initial condition and forcing functions. The main idea is that, based on the introduction of an ensemble average and a special semi-implicit time discretization, the discrete systems for the multiple flow simulations share a common coefficient matrix. Thus, instead of solving J linear system with J right-hand sides (RHS), one only need solve one linear system with J RHS. This leads to great computational saving in linear solvers when either the LU factorization (for small-scale systems) or a block iterative algorithm (for large-

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“…In a study (Guermond & Quartapelle, 1998) it is affirmed that the fractional step method based on the Poisson equation for pressure overcomes the inf-sup condition under a suitable limitation for t and for P 1 /P 1 linear approximation; without respecting the temporal limitation or for equal-order P 2 /P 2 approximation, the numerical solution is unstable. Since we are using equal-order P 2 /P 2 approximation, stabilization techniques needed in order to complete the formulation (36-41).…”

Section: The Inf-sup Condition and A Stabilizationmentioning

confidence: 99%

A cost-effective FE method for 2D Navier–Stokes equations

Jotsa

Pennati

2015

Engineering Applications of Computational Fluid Mechanics

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A cost-effective approach to the solution of 2D Navier-Stokes equations for incompressible fluid flow problems is presented. The aim is to reach a good compromise between numerical properties and computational efficiency. In order to achieve the set goal, the nonlinear convective terms are approximated by means of characteristics and spatial approximations of equal order are performed by polynomials of degree two. In this way, the computational kernels are reduced to elliptic ones for which solution very efficient techniques are available. The time-advancing is afforded by a fractional step method combined with a stabilization technique suitably simplified, so that the inf-sup condition is easily overcome. The algebraic systems generated by the new technique are solved by an iterative solver (Bi-CGSTAB), preconditioned by means of a suitable Schwarz additive scalable preconditioner. The properties of the new method have been confirmed from the comparison among the results obtained by it, and those obtained from other methods in the solution of some well known test problems. The obtained results, both in terms of accuracy and computational efficiency, make realistic the possibility to extend the method to 3D problems and to develop a multidomain approach.

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On stability and convergence of projection methods based on pressure Poisson equation

Cited by 169 publications

References 16 publications

Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces

Finite-element/level-set/operator-splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

A cost-effective FE method for 2D Navier–Stokes equations

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