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On some finite difference schemes for solution of hyperbolic heat conduction problems

Raimondas Čiegis, Aleksas Mirinavičius (2011)

Open Mathematics

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem...

On source terms and boundary conditions using arbitrary high order discontinuous Galerkin schemes

Michael Dumbser, Claus-dieter Munz (2007)

International Journal of Applied Mathematics and Computer Science

This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann...

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On Spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping∗

Bao-Zhu Guo, Guo-Dong Zhang (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we study the one-dimensional wave equation with Boltzmann damping. Two different Boltzmann integrals that represent the memory of materials are considered. The spectral properties for both cases are thoroughly analyzed. It is found that when the memory of system is counted from the infinity, the spectrum of system contains a left half complex plane, which is sharp contrast to the most results in elastic vibration systems that the vibrating dynamics can be considered from the vibration...

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

Camille Laurent (2011)

Journées Équations aux dérivées partielles

We study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard [2] on ℝ 3 , is performed by taking care of possible...

On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations

Eliane Bécache, Patrick Joly (2002)

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

In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This last...

On the analysis of Bérenger's Perfectly Matched Layers for Maxwell's equations

Eliane Bécache, Patrick Joly (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we investigate the Perfectly Matched Layers (PML) introduced by Bérenger [3] for designing efficient numerical absorbing layers in electromagnetism. We make a mathematical analysis of this model, first via a modal analysis with standard Fourier techniques, then via energy techniques. We obtain uniform in time stability results (that make precise some results known in the literature) and state some energy decay results that illustrate the absorbing properties of the model. This...

On the anomalous singularities of the solutions to some classes of weakly hyperbolic semilinear systems. Examples

Petar Popivanov, Iordan Iordanov (2003)

Banach Center Publications

This paper deals with the newly observed singularities of the solutions of some specific examples of weakly hyperbolic semilinear systems in R². Two, respectively three, characteristics are supposed to be mutually tangential at the origin only and the initial data are continuous only. The exact strength of the new-born singularities is investigated too.

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set Ω ⊂ ℝ n , with Dirichlet boundary conditions. The observation is done on a subset ω of Lebesgue measure | ω | = L | Ω | , where L ∈ ( 0 , 1 ) is fixed. We denote by 𝒰 L the class of all possible such subsets. Let T > 0 . We consider first the benchmark problem of maximizing...

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