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E-raamat: Two-Scale Approach to Oscillatory Singularly Perturbed Transport Equations

  • Formaat: EPUB+DRM
  • Sari: Lecture Notes in Mathematics 2190
  • Ilmumisaeg: 05-Oct-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319646688
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Two-Scale Approach to Oscillatory Singularly Perturbed Transport EquationsSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Lecture Notes in Mathematics 2190
  • Ilmumisaeg: 05-Oct-2017
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319646688
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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.


Arvustused

This is a good research monograph for people working on theoretical and numerical aspects of oscillatory singularly perturbed differential equations. The book is well-written with several examples from various applications. This book provides the complete picture of two-scale convergence approach for homogenization problems and the numerical approach. This monograph is excellent and well-written. This book will be very useful for mathematicians and engineers working on multiscale problems. (Srinivasan Natesan, zbMATH 1383.65084, 2018)

Part I Two-Scale Convergence
1 Introduction
3(18)
1.1 First Statements on Two-Scale Convergence
3(1)
1.2 Two-Scale Convergence and Homogenization
3(18)
1.2.1 How Homogenization Led to the Concept of Two-Scale Convergence
3(12)
1.2.2 A Remark Concerning Periodicity
15(1)
1.2.3 A Remark Concerning Weak-* Convergence
15(6)
2 Two-Scale Convergence: Definition and Results
21(14)
2.1 Background Material on Two-Scale Convergence
21(3)
2.1.1 Definitions
21(2)
2.1.2 Link with Weak Convergence
23(1)
2.2 Two-Scale Convergence Criteria
24(11)
2.2.1 Injection Lemma
24(5)
2.2.2 Two-Scale Convergence Criterion
29(1)
2.2.3 Strong Two-Scale Convergence Criterion
30(5)
3 Applications
35(56)
3.1 Homogenization of Ordinary Differential Equations
35(8)
3.1.1 Textbook Case, Setting and Asymptotic Expansion
35(4)
3.1.2 Justification of Asymptotic Expansion Using Two-Scale Convergence
39(4)
3.2 Homogenization of Oscillatory Singularly-Perturbed Ordinary Differential Equations
43(23)
3.2.1 Equation of Interest and Setting
43(2)
3.2.2 Asymptotic Expansion Results
45(2)
3.2.3 Asymptotic Expansion Calculations
47(6)
3.2.4 Justification Using Two-Scale Convergence I: Results
53(1)
3.2.5 Justification Using Two-Scale Convergence II: Proofs
54(12)
3.3 Homogenization of Hyperbolic Partial Differential Equations
66(8)
3.3.1 Textbook Case and Setting
66(1)
3.3.2 Order-0 Homogenization
66(3)
3.3.3 Order-1 Homogenization
69(5)
3.4 Homogenization of Linear Singularly-Perturbed Hyperbolic Partial Differential Equations
74(17)
3.4.1 Equation of Interest and Setting
74(1)
3.4.2 An a Priori Estimate
75(1)
3.4.3 Weak Formulation with Oscillating Test Functions
75(1)
3.4.4 Order-0 Homogenization: Constraint
76(1)
3.4.5 Order-0 Homogenization: Equation for V
76(2)
3.4.6 Order-1 Homogenization: Preparations: Equations for U and u
78(1)
3.4.7 Order-1 Homogenization: Strong Two-Scale Convergence of uE
79(1)
3.4.8 Order-1 Homogenization: The Function W1
80(2)
3.4.9 Order-1 Homogenization: A Priori Estimate and Convergence
82(1)
3.4.10 Order-1 Homogenization: Constraint
83(1)
3.4.11 Order-1 Homogenization: Equation for V1
84(3)
3.4.12 Concerning Numerics
87(4)
Part II Two-Scale Numerical Methods
4 Introduction
91(2)
5 Two-Scale Numerical Method for the Long-Term Forecast of the Drift of Objects in an Ocean with Tide and Wind
93(16)
5.1 Motivation and Model
93(3)
5.1.1 Motivation
93(1)
5.1.2 Model of Interest
94(2)
5.2 Two-Scale Asymptotic Expansion
96(4)
5.2.1 Asymptotic Expansion
96(2)
5.2.2 Discussion
98(2)
5.3 Two-Scale Numerical Method
100(9)
5.3.1 Construction of the Two-Scale Numerical Method
100(2)
5.3.2 Validation of the Two-Scale Numerical Method
102(7)
6 Two-Scale Numerical Method for the Simulation of Particle Beams in a Focussing Channel
109(12)
6.1 Some Words About Beams and the Model of Interest
109(4)
6.1.1 Beams
109(1)
6.1.2 Equations of Interest
110(1)
6.1.3 Two-Scale Convergence
111(2)
6.2 Two-Scale PIC Method
113(8)
6.2.1 Formulation of the Two-Scale Numerical Method
113(4)
6.2.2 Numerical Results
117(4)
References 121
Emmanuel Frénod is Professor of Applied Mathematics at Université Bretagne Sud.
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