On the Long-Time Stability of the Implicit Euler Scheme for the Two-Dimensional Navier--Stokes Equations

Tone, Florentina; Wirosoetisno, D.

doi:10.1137/040618527

Cited by 67 publications

(77 citation statements)

References 12 publications

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“…The new alternative treatment of the nonlinear advection term proves to be crucial in our long time stability analaysis. There is a long list of work on time and spatial discretization of the NSE and related dissipative systems that preserves the dissipativity in various forms (see for instance [10,11,15,22,23,24,25,29,34,37,38,46,55] among many others). In particular, a general framework for the convergence of global attractor (in upper semicontinous fashion) for one step scheme was derived by Hale, Lin and Raugel [21].…”

Section: 4)mentioning

confidence: 99%

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

Wang

2012

Numer. Math.

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Abstract. We investigate the long time behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box:The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme inL 2 ,Ḣ 1 per andḢ 2 per provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed.Key words. Navier-Stokes equations, invariant measures, long time statistical properties, second order in time scheme, long time global stability AMS subject classifications. . If the the long time statistics of the system, i.e., the climate, is the subject of study, we then need to investigate the invariant measures of the system since it is the invariant measure (or stationary statistical solutions) that describes the long time statistics of the underlying dynamical system. Since most of these chaotic and/or turbulent systems are not amenable to analytical techniques at the present and in the near future, the issue of the development of numerical methods that are able to capture the long time statistics becomes very important. Higher order efficient schemes are apparently preferred due to the long time integration needed.In this paper, we will focus on the development and analysis of an efficient second order two step numerical method that is able to capture the long time statistics for the following two dimensional Navier-Stokes system for homogeneous incompressible Newtonian fluids in the vorticity-streamfunction formulation (see for instance [36])

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Section: 4)mentioning

confidence: 99%

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

Wang

2012

Numer. Math.

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“…To continue our study, we first collect some results obtained in [25]. The first two concern initial data u 0 ∈ H. There exists κ 0 > 0, independent of u 0 , n, k, such that the following hold:…”

Section: Earlier Contributionsmentioning

confidence: 99%

“…We now turn our attention to initial data u 0 ∈ V and recall the main result derived in [25], tailored to our case. Theorem 3.5.…”

Section: Earlier Contributionsmentioning

confidence: 99%

“…Nonetheless, we will be able to show that, in fact, this difficulty can be overcome by defining a family of multivalued discrete semigroups {S k , 0 < k ≤ κ 1 }, where κ 1 > 0 is constant and independent of u 0 . Taking advantage of previous results contained in [25], we will address the issue of the existence of a family of discrete global attractors A k of S k and we will prove that A k → A as k → 0 in a suitable sense, where A is the global attractor of the single-valued dynamical system generated by (1.1)-(1.3).…”

Section: Introductionmentioning

confidence: 99%

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Multivalued attractors and their approximation: applications to the Navier–Stokes equations

Zelati

Tone

2012

Numer. Math.

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Abstract. This article is devoted to the study of multivalued semigroups and their asymptotic behavior, with particular attention to iterations of set-valued mappings. After developing a general abstract framework, we present an application to a time discretization of the two-dimensional Navier-Stokes equations. More precisely, we prove that the fully implicit Euler scheme generates a family of discrete multivalued dynamical systems, whose global attractors converge to the global attractor of the continuous system as the time-step parameter approaches zero.

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“…Outros estudos recentes têm sido realizados na estabilidade de esquemas implícitos aplicados às equações de Navier-Stokes: para fórmulas de diferenças atrasadas (backward dierentiation formulas) [83], para uma discretização totalmente implícita das equações de Navier-Stokes utilizando o método de Euler implícito [166], e para esquemas de Runge-Kutta [111].…”

Section: Esquemas Implícitos Na Metodologia Gensmacunclassified

Análise e implementação de métodos implícitos e de projeção para escoamentos com superfície livre.

Oishi¹

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Finalmente, e acima de tudo, a Deus pela minha vida.iii iv ResumoNo contexto do método MAC e baseado em esquemas de diferenças nitas, este trabalho apresenta três estudos: i) uma análise de estabilidade, ii) o desenvolvimento de técnicas implícitas e, iii) a construção de métodos de projeção para escoamentos com superfície livre.Na análise de estabilidade, o principal resultado mostra que o método de Crank-Nicolson torna-se condicionalmente estável quando aplicado para uma malha deslocada com a discretização explícita das condições de contorno do tipo Dirichlet. Entretanto, o mesmo método com condições de contorno implícitas é incondicionalmente estável.Para obter métodos mais estáveis, formulações implícitas são desenvolvidas para a equação da pressão na superfície livre, derivada da condição de tensão normal. Esta estratégia resulta no acoplamento dos campos de velocidade e pressão, o que exige a introdução de novos métodos de projeção.Os métodos de projeção assim desenvolvidos resultam em novas metodologias para escoamentos com superfície livre que são apropriados para o tratamento de problemas com baixo número de Reynolds. Além disso, mostra-se que os métodos propostos podem ser aplicados para uidos viscoelásticos. Novas estratégias são derivadas para obter métodos de projeção de segunda ordem de precisão para escoamentos com superfícies livres.Além dos resultados teóricos sobre a estabilidade de esquemas numéricos, técnicas implícitas e métodos de projeção, testes computacionais são realizados e comparados para consolidação da teoria apresentada. Os resultados numéricos são obtidos no sistema FREEFLOW.A eciência e robustez das técnicas desenvolvidas neste trabalho são demonstradas na solução de problemas tridimensionais complexos com superfície livre e baixo número de Reynolds, incluindo os problemas do jato oscilante e do inchamento do extrudado. In the context of the MAC method and based on nite dierence schemes, this work presents three studies: i) a stability analysis, ii) the development of implicit techniques, and iii) the construction of projection methods for free surface ows.In the stability analysis, the main result shows a precise stability restriction on the CrankNicolson method when one uses a staggered grid with Dirichlet explicit boundary conditions. However, the same method with implicit boundary conditions becomes unconditionally stable.In order to obtain more stable methods, implicit formulations are applied for the pressure equation at the free surface, which is derived from the normal stress condition. This approach results in a coupling of the velocity and pressure elds; hence new projection methods for free surface ows need to be developed.The developed projection methods result in new methodologies for low Reynolds number free surface ows. It is also shown that the proposed methods can be applied for viscoelastic uids. New strategies are derived for obtaining second-order accurate projection methods for free surface ows.In addition to the theoretical results on the stability of numeri...

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On the Long-Time Stability of the Implicit Euler Scheme for the Two-Dimensional Navier--Stokes Equations

Cited by 67 publications

References 12 publications

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

Multivalued attractors and their approximation: applications to the Navier–Stokes equations

Análise e implementação de métodos implícitos e de projeção para escoamentos com superfície livre.

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