Numerical Continuation Methods

Allgower, Eugene L.; Georg, Kurt

doi:10.1007/978-3-642-61257-2

Cited by 1,100 publications

(524 citation statements)

References 291 publications

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“…c n ε n (25) and insert these directly into (2). Equating at orders of ε one recovers, at order one, the linear solutions (6) provided that c 0 satisfies (5) for two wavenumbers k 1 and k 2 .…”

Section: Boundary Perturbation Methods For Traveling Wavesmentioning

confidence: 88%

“…The computations of Nicholls [50,51], and Craig & Nicholls [24] were based on an OE implementation of the DNO, but were performed at a relatively low perturbation order, N = 5. These simulations utilized Boundary Perturbations solely in the computation of the DNO and otherwise used numerical continuation [42,2] to find solutions of (11) along the branch using a predictorcorrector algorithm.…”

Section: For Demonstrations)mentioning

confidence: 99%

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Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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Key words Water waves, free-surface fluid flows, ideal fluid flows, boundary perturbation methods, spectral methods.The most successful equations for the modeling of ocean wave phenomena are the freesurface Euler equations. Their solutions accurately approximate a wide range of physical problems from open-ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem." Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed.

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Section: Boundary Perturbation Methods For Traveling Wavesmentioning

confidence: 88%

Section: For Demonstrations)mentioning

confidence: 99%

Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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“…These methods of computational linear algebra are widely used and their details are well-known. Therefore we are not discussing it here ( [22,23]). …”

Section: A Few Words About the Specifics Of The Computational Difficumentioning

confidence: 99%

Legendre integrators, post-processing and quasiequilibrium

Gorban

Karlin

2004

Journal of Non-Newtonian Fluid Mechanics

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A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapest way to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium ("maximum entropy", MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.

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“…The standard prediction stage is carried out by a tangent (or Euler) approximation, and the standard correction is performed by Newton's method or one of its variants. We refer to [1] for an excellent overview of continuation techniques. Now, while continuing the curve of equilibria, a robust algorithm must also be able to adaptively choose the steps to be taken, and to detect bifurcation values (e.g., points where two solution curves intersect, or points where -for (1)-the curve fails to be parametrizable in α).…”

Section: Introductionmentioning

confidence: 99%

Path Following by SVD

Dieci

Gasparo

Papini

2006

Lecture Notes in Computer Science

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Abstract. In this paper, we propose a path-following method for computing a curve of equilibria of a dynamical system, based upon the smooth Singular Value Decomposition (SVD) of the Jacobian matrix. Our method is capable of detecting fold points, and continuing past folds. It is also able to detect branch points and to switch branches at such points. Algorithmic details and examples are given.Subject Classifications: 65F15, 65F99.

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Numerical Continuation Methods

Cited by 1,100 publications

References 291 publications

Boundary Perturbation Methods for Water Waves

Boundary Perturbation Methods for Water Waves

Legendre integrators, post-processing and quasiequilibrium

Path Following by SVD

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