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Numerical computation of solitons for optical systems

Laurent Di Menza (2009)

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

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ . In a second part, we compute...

Numerical computation of solitons for optical systems

Laurent Di Menza (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ. In a second part, we compute...

Numerical homogenization: survey, new results, and perspectives

Antoine Gloria (2012)

ESAIM: Proceedings

These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which H-converges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the...

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

Milena Dimova, Stefka Dimova, Daniela Vasileva (2013)

Open Mathematics

The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach...

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2003)

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

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Numerical resolution of an “unbalanced” mass transport problem

Jean-David Benamou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a modification of the Monge–Kantorovitch problem of exponent 2 which accommodates non balanced initial and final densities. The augmented Lagrangian numerical method introduced in [6] is adapted to this “unbalanced” problem. We illustrate the usability of this method on an idealized error estimation problem in meteorology.

Numerical simulations for nodal domains and spectral minimal partitions

Virginie Bonnaillie-Noël, Bernard Helffer, Gregory Vial (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We recall here some theoretical results of Helffer et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire (2007) doi:10.1016/j.anihpc.2007.07.004] about minimal partitions and propose numerical computations to check some of their published or unpublished conjectures and exhibit new ones.

Numerical solution of a 1-d elastohydrodynamic problem in magnetic storage devices

Iñigo Arregui, José Jesús Cendán, Carlos Parés, Carlos Vázquez (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a...

Numerical solution of several models of internal transonic flow

Jaroslav Fořt, Karel Kozel (2003)

Applications of Mathematics

The paper deals with numerical solution of internal flow problems. It mentions a long tradition of mathematical modeling of internal flow, especially transonic flow at our department. Several models of flow based on potential equation, Euler equations, Navier-Stokes and Reynolds averaged Navier-Stokes equations with proper closure are considered. Some mathematical and numerical properties of the model are mentioned and numerical results achieved by in-house developed methods are presented.

Numerical study of a new global minimizer for the Mumford-Shah functional in R3

Benoît Merlet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in 𝐑 3 . The singular set of such a new minimizer belongs to a three parameters family of sets ( 0 < δ 1 , δ 2 , δ 3 < π ) . We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of 𝐒 2 with three reentrant corners. The necessary conditions are...

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Runchang Lin, Zhimin Zhang (2009)

Applications of Mathematics

Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.

Numerical study of two sparse AMG-methods

Janne Martikainen (2003)

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

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

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