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Numerical comparisons of two long-wave limit models

Stéphane Labbé, Lionel Paumond (2004)

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

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...

Numerical comparisons of two long-wave limit models

Stéphane Labbé, Lionel Paumond (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...

Numerical controllability of the wave equation through primal methods and Carleman estimates

Nicolae Cîndea, Enrique Fernández-Cara, Arnaud Münch (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null...

Numerical evidence of nonuniqueness in the evolution of vortex sheets

Milton C. Lopes Filho, John Lowengrub, Helena J. Nussenzveig Lopes, Yuxi Zheng (2006)

ESAIM: Mathematical Modelling and Numerical Analysis


We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary...

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 methods for fourth order nonlinear degenerate diffusion problems

Jürgen Becker, Günther Grün, Martin Lenz, Martin Rumpf (2002)

Applications of Mathematics

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant),...

Numerical methods for phase transition problems

Claudio Verdi (1998)

Bollettino dell'Unione Matematica Italiana

Nel presente articolo si illustrano alcuni dei principali metodi numerici per l'approssimazione di modelli matematici legati ai fenomeni di transizione di fase. Per semplificare e contenere l'esposizione ci siamo limitati a discutere con un certo dettaglio i metodi più recenti, presentandoli nel caso di problemi modello, quali il classico problema di Stefan e l'evoluzione di superficie per curvatura media, solo accennando alle applicazioni e modelli più generali.

Numerical modeling of heat exchange and unsaturated-saturated flow in porous media

Kačur, Jozef, Mihala, Patrik, Tóth, Michal (2017)

Proceedings of Equadiff 14

We discuss the numerical modeling of heat exchange between the infiltrated water and porous media matrix. An unsaturated-saturated flow is considered with boundary conditions reflecting the external driven forces. The developed numerical method is efficient and can be used for solving the inverse problems concerning determination of transmission coefficients for heat energy exchange inside and also on the boundary of porous media. Numerical experiments support our method.

Numerical Modelling of Contact Elastic-Plastic Flows

N. M. Bessonov, S. F. Golovashchenko, V. A. Volpert (2009)

Mathematical Modelling of Natural Phenomena

Wilkins' method has been successfully used since early 60s for numerical simulation of high velocity contact elastic-plastic flows. The present work proposes some effective modifications of this method including more sophisticated material model including the Baushinger effect; modification of the algorithm based on correction of the initial configuration of a solid; a contact algorithm based on the idea of a mild contact.

Numerical modelling of flow in lower urinary tract using high-resolution methods

Brandner, Marek, Egermaier, Jiří, Kopincová, Hana, Rosenberg, Josef (2013)

Programs and Algorithms of Numerical Mathematics

We propose a new numerical scheme based on the finite volumes to simulate the urethra flow based on hyperbolic balance law. Our approach is based on the Riemann solver designed for the augmented quasilinear homogeneous formulation. The scheme has general semidiscrete wave–propagation form and can be extended to arbitrary high order accuracy. The first goal is to construct the scheme, which is well balanced, i.e. maintains not only some special steady states but all steady states which can occur....

Numerical schemes for a three component Cahn-Hilliard model

Franck Boyer, Sebastian Minjeaud (2011)

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

In this article, we investigate numerical schemes for solving a three component Cahn-Hilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by...

Numerical schemes for a three component Cahn-Hilliard model

Franck Boyer, Sebastian Minjeaud (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

In this article, we investigate numerical schemes for solving a three component Cahn-Hilliard model. The space discretization is performed by using a Galerkin formulation and the finite element method. Concerning the time discretization, the main difficulty is to write a scheme ensuring, at the discrete level, the decrease of the free energy and thus the stability of the method. We study three different schemes and prove existence and convergence theorems. Theoretical results are illustrated by...

Numerical simulation of a point-source initiated flame ball with heat losses

Jacques Audounet, Jean-Michel Roquejoffre, Hélène Rouzaud (2002)

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

This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.

Numerical simulation of a point-source initiated flame ball with heat losses

Jacques Audounet, Jean-Michel Roquejoffre, Hélène Rouzaud (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This article is devoted to the numerical study of a flame ball model, derived by Joulin, which obeys to a singular integro-differential equation. The numerical scheme that we analyze here, is based upon a one step method, and we are interested in its long-time behaviour. We recover the same dynamics as in the continuous case: quenching, or stabilization of the flame, depending on heat losses, and an energy input parameter.

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