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Über dieses Buch

This book presents the classical results of the two-scale convergence theory and explains – using several figures – why it works. It then shows how to use this theory to homogenize ordinary differential equations with oscillating coefficients as well as oscillatory singularly perturbed ordinary differential equations. In addition, it explores the homogenization of hyperbolic partial differential equations with oscillating coefficients and linear oscillatory singularly perturbed hyperbolic partial differential equations. Further, it introduces readers to the two-scale numerical methods that can be built from the previous approaches to solve oscillatory singularly perturbed transport equations (ODE and hyperbolic PDE) and demonstrates how they can be used efficiently. This book appeals to master’s and PhD students interested in homogenization and numerics, as well as to the Iter community.



Two-Scale Convergence


Chapter 1. Introduction

The concept of Two-Scale Convergence was introduced in two papers of Nguetseng
Emmanuel Frénod

Chapter 2. Two-Scale Convergence: Definition and Results

There are several variants of the main two-scale convergence result, more or less well adapted to targeted applications and involving various functional spaces (see Nguetseng [49, 50], Allaire [4], Amar [6], Casado-Díaz and Gayte [15], Frénod et al. [35], Nguetseng and Woukeng [52], and Nguetseng and Svanstedt [51]).
Emmanuel Frénod

Chapter 3. Applications

Before entering the core of our targeted applications, which concerns strong oscillations in transport phenomena, I will show on a simple ordinary differential equation that involves oscillations how two-scale convergence can be used.
Emmanuel Frénod

Two-Scale Numerical Methods


Chapter 4. Introduction

Regradless of which phenomena involving high-frequency oscillations or heterogeneities one is dealing with, their simulations all face.
Emmanuel Frénod

Chapter 5. Two-Scale Numerical Method for the Long-Term Forecast of the Drift of Objects in an Ocean with Tide and Wind

The drift of objects in the coastal ocean waters is potentially dangerous for human activities and marine ecosystems. For instance, drifting containers may cause serious accidents in the event of collision with ships, oil spills may have very negative impact especially in coastal areas, etc.
Emmanuel Frénod

Chapter 6. Two-Scale Numerical Method for the Simulation of Particle Beams in a Focussing Channel

A beam is a set of charged particles, which all have the same charge, all move in the same direction, the so-called the beam axis, and are confined around this axis.
Emmanuel Frénod


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