Gradient stability of high-order BDF methods and some applications

Bouchriti, Anass; Pierre, Morgan; Alaa, Nour Eddine

doi:10.1080/10236198.2019.1709062

Cited by 8 publications

(17 citation statements)

References 25 publications

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“…This situation can be observed when the distancing measures has been applied differently with respect to various regions. However, this generalized model is much more complicated problem and cannot be solved directly by our algorithm, for that in the future work we intend to analyse mathematically this model and also solve numerically this problem by using our algorithm combined with the BDF scheme developed recently in (Bouchriti et al 2020).…”

Section: Discussionmentioning

confidence: 99%

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

Gounane¹,

Barkouch²,

Atlas

et al. 2021

Epidemiologic Methods

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Recently, various mathematical models have been proposed to model COVID-19 outbreak. These models are an effective tool to study the mechanisms of coronavirus spreading and to predict the future course of COVID-19 disease. They are also used to evaluate strategies to control this pandemic. Generally, SIR compartmental models are appropriate for understanding and predicting the dynamics of infectious diseases like COVID-19. The classical SIR model is initially introduced by Kermack and McKendrick (cf. (Anderson, R. M. 1991. “Discussion: the Kermack–McKendrick Epidemic Threshold Theorem.” Bulletin of Mathematical Biology 53 (1): 3–32; Kermack, W. O., and A. G. McKendrick. 1927. “A Contribution to the Mathematical Theory of Epidemics.” Proceedings of the Royal Society 115 (772): 700–21)) to describe the evolution of the susceptible, infected and recovered compartment. Focused on the impact of public policies designed to contain this pandemic, we develop a new nonlinear SIR epidemic problem modeling the spreading of coronavirus under the effect of a social distancing induced by the government measures to stop coronavirus spreading. To find the parameters adopted for each country (for e.g. Germany, Spain, Italy, France, Algeria and Morocco) we fit the proposed model with respect to the actual real data. We also evaluate the government measures in each country with respect to the evolution of the pandemic. Our numerical simulations can be used to provide an effective tool for predicting the spread of the disease.

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Section: Discussionmentioning

confidence: 99%

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

Gounane¹,

Barkouch²,

Atlas

et al. 2021

Epidemiologic Methods

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“…They are much more sharper than the recent results in [4, Theorems 3.2 and 3.6] with the eigenvalue estimates σ L4 = 41/72 ≈ 0.5694 and σ L5 = 0.1 for the BDF-4 and BDF-5 formulas, respectively. The present treatment gives the explicit expressions of the Lyapunov functionals G k and is quite different from the recent techniques in [4,24].…”

Section: Lemma 23 For the Real Sequence {Vmentioning

confidence: 93%

“…The discrete energy dissipation law and the concise L 2 norm error estimate were established under a practical step-ratio constraint. This lower order scheme would be well suited for the fast varying solutions especially in the early coarsening process [29]; while highorder stable methods should be more preferable for slowly varying solutions during the long-time process approaching the steady state [4,14,15,24]. Nonetheless, due to the lack of proper discrete energy techniques, the stability and convergence of the non-A-stable BDF-k (k = 3, 4, 5, 6) schemes for nonlinear phase field models have not been well studied in the literatures.…”

Section: Introductionmentioning

confidence: 99%

“…We focus on the intrinsic energy stability properties of the non-A-stable BDF-k formulas themselves, that is, some positive constants σ Lk (the larger, the better), two nonnegative quadratic functionals G k and R k are sought such that the BDF-k kernels b (k) j defined in (1.5) satisfy the following discrete gradient structure, cf. [4,24] and [26,Section 5.6],…”

Section: Introductionmentioning

confidence: 99%

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Discrete gradient structures of BDF methods up to fifth-order for the phase field crystal model

Liao¹,

Kang²

2022

Preprint

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The well-known backward difference formulas (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-k (k = 3, 4, 5) formulas, we establish the energy dissipation laws at the discrete levels and then obtain the priori solution estimates for the associated numerical schemes (however, we can not build any discrete energy dissipation law for the corresponding BDF-6 scheme because the BDF-6 formula itself does not have any discrete gradient structures). With the help of the discrete orthogonal convolution kernels and Young-type convolution inequalities, some concise L 2 norm error estimates are established via the discrete energy technique. To the best of our knowledge, this is the first time such type L 2 norm error estimates of non-A-stable BDF schemes are obtained for nonlinear parabolic equations. Numerical examples are presented to verify and support the theoretical analysis.

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“…Finite element methods [43,53,60], finite difference methods [23,24,28,30,55], spectral-Galerkin approaches [127] and discontinuous Galerkin methods [9,139] with such properties are available. Regarding the time semidiscretization, we refer the reader to the reviews [126,127,133] and to the paper [21].…”

Section: Matthieu Brachet Philippe Parnaudeau and Morgan Pierrementioning

confidence: 99%

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

Brachet

Parnaudeau

Pierre

2022

DCDS-S

Self Cite

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<p style='text-indent:20px;'>We review space and time discretizations of the Cahn-Hilliard equation which are energy stable. In many cases, we prove that a solution converges to a steady state as time goes to infinity. The proof is based on Lyapunov theory and on a Lojasiewicz type inequality. In a few cases, the convergence result is only partial and this raises some interesting questions. Numerical simulations in two and three space dimensions illustrate the theoretical results. Several perspectives are discussed.</p>

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Gradient stability of high-order BDF methods and some applications

Cited by 8 publications

References 25 publications

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

Discrete gradient structures of BDF methods up to fifth-order for the phase field crystal model

Convergence to equilibrium for time and space discretizations of the Cahn-Hilliard equation

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