Discrete variational derivative method : a structure-preserving numerical method for partial differential equations

Title:

Personal Author:

Furihata, Daisuke

Series:

Chapman and Hall/CRC numerical analysis and scientific computation series

Publication Information:

Boca Raton, FL : Chapman and Hall/CRC, 2010

Physical Description:

xi, 376 p. : ill. ; 24 cm.

ISBN:

9781420094459

Abstract:

"Many important problems in engineering and science are modeled by nonlinear partial differential equations (PDEs). A new trend in PDEs, called structure-preserving numerical methods, has recently developed. This book is devoted to one such technique, called the discrete variational derivative method. First, the text introduces the key factors and the basic ideas of this method, followed by target problems solvable by the method. The second section describes the rigorous mathematics in detail along with relevant applications, which are illustrated by worked examples. It concludes with a comprehensive of listing of essential references on structure-preserving algorithms for advanced readers"-- Provided by publisher.

Subject Term:

Differential equations, Partial -- Numerical solutions

Nonlinear theories

Engineering mathematics

Added Author:

Matsuo, Takayasu

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010251469	QA377 F87 2011	Open Access Book	Book	Searching... Unknown

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Summary

Nonlinear Partial Differential Equations (PDEs) have become increasingly important in the description of physical phenomena. Unlike Ordinary Differential Equations, PDEs can be used to effectively model multidimensional systems.

The methods put forward in Discrete Variational Derivative Method concentrate on a new class of "structure-preserving numerical equations" which improves the qualitative behaviour of the PDE solutions and allows for stable computing. The authors have also taken care to present their methods in an accessible manner, which means that the book will be useful to engineers and physicists with a basic knowledge of numerical analysis. Topics discussed include:

"Conservative" equations such as the Korteweg-de Vries equation (shallow water waves) and the nonlinear Schrödinger equation (optical waves) "Dissipative" equations such as the Cahn-Hilliard equation (some phase separation phenomena) and the Newell-Whitehead equation (two-dimensional Bénard convection flow) Design of spatially and temporally high-order schemas Design of linearly-implicit schemas Solving systems of nonlinear equations using numerical Newton method libraries

Preface	p. ix
1 Introduction and Summary	p. 1
1.1 An Introductory Example: Spinodal Decomposition	p. 1
1.2 History	p. 10
1.3 Derivation of Dissipative or Conservative Schemes	p. 12
1.3.1 Procedure for First-Order Real-Valued PDEs	p. 12
1.3.2 Procedure for First-Order Complex-Valued PDEs	p. 19
1.3.3 Procedure for Systems of First-Order PDEs	p. 24
1.3.4 Procedure for Second-Order PDEs	p. 27
1.4 Advanced Topics	p. 34
1.4.1 Design of Higher-Order Schemes	p. 34
1.4.2 Design of Linearly Implicit Schemes	p. 40
1.4.3 Further Remarks	p. 47
2 Target Partial Differential Equations	p. 49
2.1 Variational Derivatives	p. 49
2.2 First-Order Real-Valued PDEs	p. 52
2.3 First-Order Complex-Valued PDEs	p. 58
2.4 Systems of First-Order PDEs	p. 60
2.5 Second-Order PDEs	p. 65
3 Discrete Variational Derivative Method	p. 69
3.1 Discrete Symbols and Formulas	p. 69
3.2 Procedure for First-Order Real-Valued PDEs	p. 75
3.2.1 Discrete Variational Derivative: Real-Valued Case	p. 75
3.2.2 Design of Schemes	p. 80
3.2.3 User's Choices	p. 87
3.3 Procedure for First-Order Complex-Valued PDEs	p. 93
3.3.1 Discrete Variational Derivative: Complex-Valued Case	p. 93
3.3.2 Design of Schemes	p. 96
3.4 Procedure for Systems of First-Order PDEs	p. 101
3.4.1 Design of Schemes	p. 105
3.5 Procedure for Second-Order PDEs	p. 110
3.5.1 First Approach: Direct Variation	p. 111
3.5.2 Second Approach: System of PDEs	p. 115
3.6 Preliminaries on Discrete Functional Analysis	p. 119
3.6.1 Discrete Function Spaces	p. 119
3.6.2 Discrete Inequalities	p. 121
3.6.3 Discrete Gronwall Lemma	p. 126
4 Applications	p. 129
4.1 Target PDEs 1	p. 129
4.1.1 Cahn-Hilliard Equation	p. 129
4.1.2 Allen-Cahn Equation	p. 149
4.1.3 Fisher-Kolmogorov Equation	p. 153
4.2 Target PDEs 2	p. 155
4.2.1 Korteweg-de Vries Equation	p. 157
4.2.2 Zakharov-Kuznetsov Equation	p. 159
4.3 Target PDEs 3	p. 164
4.3.1 Complex-Valued Ginzburg-Landau Equation	p. 164
4.3.2 Newell-Whitehead Equation	p. 165
4.4 Target PDEs 4	p. 167
4.4.1 Nonlinear Schrödinger Equation	p. 167
4.4.2 Gross-Pitaevskii Equation	p. 180
4.5 Target PDEs 5	p. 182
4.5.1 Zakharov Equations	p. 183
4.6 Target PDEs 7	p. 185
4.6.1 Nonlinear Klein-Gordon Equation	p. 185
4.6.2 Shimoji-Kawai Equation	p. 189
4.7 Other Equations	p. 191
4.7.1 Keller-Segel Equation	p. 191
4.7.2 Camassa-Holm Equation	p. 195
4.7.3 Benjamin-Bona-Mahony Equation	p. 212
4.7.4 Feng Equation	p. 222
5 Advanced Topic I: Design of High-Order Schemes	p. 227
5.1 Orders of Accuracy of Schemes	p. 227
5.2 Spatially High-Order Schemes	p. 229
5.2.1 Discrete Symbols and Formulas	p. 229
5.2.2 Discrete Variational Derivative	p. 231
5.2.3 Design of Schemes	p. 233
5.2.4 Application Examples	p. 238
5.3 Temporally High-Order Schemes: Composition Method	p. 247
5.4 Temporally High-Order Schemes: High-Order Discrete Varia-tional Derivatives	p. 248
5.4.1 Discrete Symbols	p. 249
5.4.2 Central Idea for High-Order Discrete Derivative	p. 250
5.4.3 Temporally High-Order Discrete Variational Derivative and Design of Schemes	p. 251
6 Advanced Topic II: Design of Linearly Implicit Schemes	p. 271
6.1 Basic Idea for Constructing Linearly Implicit Schemes	p. 271
6.2 Multiple-Points Discrete Variational Derivative	p. 274
6.2.1 For Real-Valued PDEs	p. 274
6.2.2 For Complex-Valued PDEs	p. 275
6.3 Design of Schemes	p. 277
6.3.1 For Real-Valued PDEs	p. 277
6.3.2 For Complex-Valued PDEs	p. 279
6.4 Applications	p. 280
6.4.1 Cahn-Hilliard Equation	p. 280
6.4.2 Odd-Order Nonlinear Schrödinger Equation	p. 283
6.4.3 Ginzburg-Landau Equation	p. 283
6.4.4 Zakharov Equations	p. 284
6.4.5 Newell-Whitehead Equation	p. 285
6.5 Remarks on the Stability of Linearly Implicit Schemes	p. 288
7 Advanced Topic III: Further Remarks	p. 293
7.1 Solving System-of Nonlinear Equations	p. 293
7.1.1 Use of Numerical Newton Method Libraries	p. 294
7.1.2 Variants of Newton Method	p. 295
7.1.3 Spectral Residual Methods	p. 296
7.1.4 Implementation as a Predictor-Corrector Method	p. 298
7.2 Switch to Galerkin Framework	p. 298
7.2.1 Design of Galerkin Schemes	p. 299
7.2.2 Application Examples	p. 309
7.3 Extension to Non-Rectangular Meshes on 2D Region	p. 348
Appendix A: Semi-Discrete Schemes' in Space	p. 353
Appendix B: Proof of Proposition 3.4	p. 357
Bibliography	p. 359
Index	p. 373

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