Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations

• Main Author: Frénod, Emmanuel (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Cham : Springer International Publishing : Imprint: Springer, 2017.
• Series: Lecture Notes in Mathematics, 2190
• Subjects: Mathematics.; Numerical analysis.; Numerical Analysis.

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 diff...

Table of Contents:

• I Two-Scale Convergence.- 1 Introduction.- 1.1 First Statements on Two-Scale Convergence.- 1.2 Two-Scale Convergence and Homogenization.- 1.2.1 How Homogenization Led to the Concept of Two-Scale Convergence.- 1.2.2 A Remark Concerning Periodicity.- 1.2.3 A Remark Concerning Weak-* Convergence.- 2 Two-Scale Convergence - Definition and Results.- 2.1 Background Material on Two-Scale Convergence.- 2.1.1 Definitions.- 2.1.2 Link with Weak Convergence.- 2.2 Two-Scale Convergence Criteria.- 2.2.1 Injection Lemma.- 2.2.2 Two-Scale Convergence Criterion.- 2.2.3 Strong Two-Scale Convergence Criterion.- 3 Applications.- 3.1 Homogenization of ODE.- 3.1.1 Textbook Case, Setting and Asymptotic Expansion.- 3.1.2 Justification of Asymptotic Expansion Using Two-Scale Convergence.- 3.2 Homogenization of Singularly-Perturbed ODE.- 3.2.1 Equation of Interest and Setting.- 3.2.2 Asymptotic Expansion Results.- 3.2.3 Asymptotic Expansion Calculations.- 3.2.4 Justification Using Two-Scale Convergence I: Results.- 3.2.5 Justification Using Two-Scale Convergence II: Proofs.- 3.3 Homogenization of Hyperbolic PDE.- 3.3.1 Textbook Case and Setting.- 3.3.2 Order-0 Homogenization.- 3.3.3 Order-1 Homogenization.- 3.4 Homogenization of Singularly-Perturbed Hyperbolic PDE
• 3.4.1 Equation of Interest and Setting
• 3.4.2 An a Priori Estimate
• 3.4.3 Weak Formulation with Oscillating Test Functions
• 3.4.4 Order-0 Homogenization - Constraint
• 3.4.5 Order-0 Homogenization - Equation for V
• 3.4.6 Order-1 Homogenization - Preparations: Equations for U and u
• 3.4.7 Order-1 Homogenization - Strong Two-Scale Convergence of u"
• 3.4.8 Order-1 Homogenization - The Function W1
• 3.4.9 Order-1 Homogenization - A Priori Estimate and Convergence
• 3.4.10 Order-1 Homogenization - Constraint
• 3.4.11 Order-1 Homogenization - Equation for V1
• 3.4.12 Concerning Numerics.- II Two-Scale Numerical Methods
• 4 Introduction
• 5 Two-Scale Method for Object Drift with Tide
• 5.1 Motivation and Model
• 5.1.1 Motivation
• 5.1.2 Model of Interest
• 5.2 Two-Scale Asymptotic Expansion
• 5.2.1 Asymptotic Expansion
• 5.2.2 Discussion
• 5.3 Two-Scale Numerical Method
• 5.3.1 Construction of the Two-Scale Numerical Method
• 5.3.2 Validation of the Two-Scale Numerical Method
• 6 Two-Scale Method for Beam
• 6.1 Some Words About Beams and Model of Interest
• 6.1.1 Beams
• 6.1.2 Equations of Interest
• 6.1.3 Two-Scale Convergence
• 6.2 Two-Scale PIC Method
• 6.2.1 Formulation of the Two-Scale Numerical Method
• 6.2.2 Numerical Results.