Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I

Cyranka, Jacek

doi:10.1007/s10915-013-9749-1

Cited by 18 publications

(37 citation statements)

References 27 publications

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“…For example the successful proof of the main result required performing of thousands of numerical integration steps. To address this problem we have developed an efficient algorithm for rigorous integration of dissipative PDEs forward in time [11]. Its highlight Figure 2.…”

Section: Framework For Proving Constructively Connecting Orbits In Pamentioning

confidence: 99%

“…The Lohner algorithm is presented in detail in [25,26]. We specifically use the algorithm introduced in [11], which allows for the efficient implementation of the Taylor method for differential equations in combination with the variational equations arising in partial differential equations. This algorithm combines automatic differentiation with the fast Fourier transform algorithm.…”

Section: Rigorous Integration Algorithmmentioning

confidence: 99%

“…While we do not present the details, the result is collected in the following remark. An example analysis can be found in [11].…”

Section: Remark 52 (Recursion For Normalized Derivatives) For An Ormentioning

confidence: 99%

“…Our method is general. To achieve the presented goal we develop a framework combining efficient algorithms and implementation in the C++ programming language performed by the first author [11] and the theory of cone conditions and self-consistent bounds developed by Zgliczyński et al [2,14,15,59,60,61,63]. We remark that the software is not hard coded for the particular equation studied in this paper, but is rather a generic tool that can be easily adapted for study of any dissipative PDE on a one dimensional domain with a polynomial nonlinearity and periodic/Dirichlet/Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Cyranka¹,

Wanner²

2018

SIAM J. Appl. Dyn. Syst.

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We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers.The proof of the above statement is conceptually simple, and combines several techniques from some of the authors' and Zgliczyński's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values.

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Section: Framework For Proving Constructively Connecting Orbits In Pamentioning

confidence: 99%

Section: Rigorous Integration Algorithmmentioning

confidence: 99%

“…While we do not present the details, the result is collected in the following remark. An example analysis can be found in [11].…”

Section: Remark 52 (Recursion For Normalized Derivatives) For An Ormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Cyranka¹,

Wanner²

2018

SIAM J. Appl. Dyn. Syst.

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“…The consequent works of Zgliczyński [14,15] and of Cyranka [23] are aimed at validating and studying time-periodic solutions of dissipative PDEs and time integration. These techniques can be applied to the destabilized Kuramoto-Sivashinsky equation (1) and other equations as well.…”

Section: Future Directionsmentioning

confidence: 99%

Fixed points of a destabilized Kuramoto–Sivashinsky equation

Bartha

Tucker

2015

Applied Mathematics and Computation

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Rigorous numerics for nonlinear heat equations in the complex plane of time

et al. 2022

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In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explicit bounds for the inclusion of the solution of the Cauchy problem, which is rewritten as a zero-finding problem on a certain Banach space. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov–Perron method for calculating part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

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Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I

Cited by 18 publications

References 27 publications

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Fixed points of a destabilized Kuramoto–Sivashinsky equation

Rigorous numerics for nonlinear heat equations in the complex plane of time

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