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P-adaptive Hermite methods for initial value problems∗

Ronald Chen, Thomas Hagstrom (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...

P-adaptive Hermite methods for initial value problems

Ronald Chen, Thomas Hagstrom (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...

P-adaptive Hermite methods for initial value problems∗

Ronald Chen, Thomas Hagstrom (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems...

Periodic solutions of nonlinear wave equations with non-monotone forcing terms

Massimiliano Berti, Luca Biasco (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Existence and regularity of periodic solutions of nonlinear, completely resonant, forced wave equations is proved for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. The corresponding infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. This difficulty is overcome finding a-priori estimates for the constrained minimizers of the reduced action functional, through techniques inspired by regularity theory...

Perte de régularité pour les équations d’ondes sur-critiques

Gilles Lebeau (2005)

Bulletin de la Société Mathématique de France

On prouve que le problème de Cauchy local pour l’équation d’onde sur-critique dans ℝ d , □ u + u p = 0 , p impair, avec d ≥ 3 et p > ( d + 2 ) / ( d - 2 ) , est mal posé dans H σ pour tout σ ∈ ] 1 , σ crit [ , où σ crit = d / 2 - 2 / ( p - 1 ) est l’exposant critique.

Poincaré-invariant structures in the solution manifold of a nonlinear wave equation.

Irving E. Segal (1986)

Revista Matemática Iberoamericana

The solution manifold M of the equation ⎯φ + gφ3 = 0 in Minkowski space is studied from the standpoint of the establishment of differential-geometric structures therein. It is shown that there is an almost Kähler structure globally defined on M that is Poincaré invariant. In the vanishing curvature case g = 0 the structure obtained coincides with the complex Hilbert structure in the solution manifold of the real wave equation. The proofs are based on the transfer of the equation to an ambient universal...

Polyhomogeneous solutions of wave equations in the radiation regime

Piotr T. Chruściel, Olivier Lengard (2000)

Journées équations aux dérivées partielles

While the physical properties of the gravitational field in the radiation regime are reasonably well understood, several mathematical questions remain unanswered. The question here is that of existence and properties of gravitational fields with asymptotic behavior compatible with existence of gravitational radiation. A framework to study those questions has been proposed by R. Penrose (R. Penrose, “Zero rest-mass fields including gravitation”, Proc. Roy. Soc. London A284 (1965), 159-203), and developed...

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