Diffusive N-Waves and Metastability in the Burgers Equation

Kim, Yong Jung; Tzavaras, Athanasios E.

doi:10.1137/s0036141000380516

Cited by 44 publications

(64 citation statements)

References 17 publications

(28 reference statements)

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“…These calculations follow closely those of [KT01,§5]. Using equation (2.15) and the fact that M = (q − p)/2, we see that…”

Section: Invariant Manifoldssupporting

confidence: 82%

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Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

Beck¹,

Wayne²

2011

SIAM Rev.

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The large-time behavior of solutions to Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted L 2 space, the selfsimilar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this "metastable" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.

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“…These calculations follow closely those of [KT01,§5]. Using equation (2.15) and the fact that M = (q − p)/2, we see that…”

Section: Invariant Manifoldssupporting

confidence: 82%

“…In [KT01], Kim and Tzavaras prove that the inviscid N-wave is the point-wise limit, as µ → 0, of the diffusive N-wave. Here we extend their argument to show that one also has convergence in the L 2 (m) norm.…”

Section: Proof Of Lemma 21mentioning

confidence: 99%

Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

Beck¹,

Wayne²

2011

SIAM Rev.

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“…The self-similar profiles are then diffusive, smooth and of constant sign. Nevertheless, when ν is sufficiently small and time is large (but not enough for the viscosity to be dominant), the behavior of the solutions is close to the hyperbolic case [29].…”

Section: Introductionmentioning

confidence: 86%

“…In the continuous case (see Fig. 1), the changing sign N-waves are the asymptotic profiles as t → ∞ if ν = 0 [32] and intermediate metastable states if ν > 0 [29]. In the hyperbolic regime, the key point in the identification of the asymptotic N-wave, which belongs to a two-parameter family, is the preservation of the quantities…”

Section: Discretization Schemes and Large-time Behaviormentioning

confidence: 98%

Numerical aspects of large-time optimal control of Burgers equation

2016

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Abstract.In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical approximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since numerical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax−Friedrichs scheme (MLF) and the Engquist−Osher (EO) one. It is by now wellknown that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT).Mathematics Subject Classification. 49M, 35Q35, 65M06.

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“…Kasu biskosoko soluzioek antzeko portaera dute hasiera batean (ν zenbat eta txikiagoa izan, orduan eta luzaroago iraungo du [17]), baina azkenik Gaussen kurbaren itxura duen uhinerantz jotzen dute beti. Kasu honetan, parametro bat baino ez da konstante mantentzen: Lehenengoa kurba urdinaren antzekoa izango da eta bigarrena, ostera, kurba gorria bezalakoa.…”

Section: Gai Ez-linealaunclassified

Zenbakizko biskositateak eragindako soinu-eztandaren simulazioaren zailtasunak

Pozo¹

2018

EKAIA

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Laburpena:Hegazkin supersonikoek eragindako soinu-eztandaren hedapena eremu hurbiletik lurreraino zehatz-mehatz simulatzea ez da erraza. Ereduaren denbora-tarte handietarako propietateak babesten dituzten zenbakizko metodoak behar dira. Artikulu honetan, soinu-eztandaren minimizazioarekin erlazionaturik dagoen optimizazio eta alderantzizko diseinu-problema ikertu dugu. Bereziki, bere soluzioaren hurbilketen eraginkortasuna mantentzeko gakoak ematen ditugu.Hitz gakoak: Soinu-eztanda, denbora-tarte handiko portaera, biskositatea, zenbakizko metodoak. Abstract:Simulating the propagation of the sonic-boom precisely from the near-field of a supersonic airplane down to the ground level is not easy. It requires numerical schemes that preserve the large-time properties of the model. In this article we deal with an optimization and inverse design problem related to the sonic-boom minimization. In particular, we emphasize the key points that need to be handled in order to keep the efficiency of the approximations of its solution.

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Diffusive N-Waves and Metastability in the Burgers Equation

Cited by 44 publications

References 17 publications

Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation with Small Viscosity

Numerical aspects of large-time optimal control of Burgers equation

Zenbakizko biskositateak eragindako soinu-eztandaren simulazioaren zailtasunak

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