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Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations [Kõva köide]

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(University of Tokyo, Bunkyo, Japan), (Osaka University, Toyonaka, Japan)
Teised raamatud teemal:
Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential EquationsSuurem pilt
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781420094459.html
Märksõnad:
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 Kortewegde Vries equation (shallow water waves) and the nonlinear Schrödinger equation (optical waves) "Dissipative" equations such as the CahnHilliard 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

Arvustused

The authors introduce a new class of structure preserving numerical methods which improve the qualitative behavior of solutions of partial differential equations and allow stable computing. This book should be useful to engineers and physicists with a basic knowledge of numerical analysis. Rémi Vaillancourt, Mathematical Reviews, Issue 2011m

Preface ix
1 Introduction and Summary 1(48)
1.1 An Introductory Example: Spinodal Decomposition
1(9)
1.2 History
10(2)
1.3 Derivation of Dissipative or Conservative Schemes
12(22)
1.3.1 Procedure for First-Order Real-Valued PDEs
12(7)
1.3.2 Procedure for First-Order Complex-Valued PDEs
19(5)
1.3.3 Procedure for Systems of First-Order PDEs
24(3)
1.3.4 Procedure for Second-Order PDEs
27(7)
1.4 Advanced Topics
34(15)
1.4.1 Design of Higher-Order Schemes
34(6)
1.4.2 Design of Linearly Implicit Schemes
40(7)
1.4.3 Further Remarks
47(2)
2 Target Partial Differential Equations 49(20)
2.1 Variational Derivatives
49(3)
2.2 First-Order Real-Valued PDEs
52(6)
2.3 First-Order Complex-Valued PDEs
58(2)
2.4 Systems of First-Order PDEs
60(5)
2.5 Second-Order PDEs
65(4)
3 Discrete Variational Derivative Method 69(60)
3.1 Discrete Symbols and Formulas
69(6)
3.2 Procedure for First-Order Real-Valued PDEs
75(18)
3.2.1 Discrete Variational Derivative: Real-Valued Case
75(5)
3.2.2 Design of Schemes
80(7)
3.2.3 User's Choices
87(6)
3.3 Procedure for First-Order Complex-Valued PDEs
93(8)
3.3.1 Discrete Variational Derivative: Complex-Valued Case
93(3)
3.3.2 Design of Schemes
96(5)
3.4 Procedure for Systems of First-Order PDEs
101(9)
3.4.1 Design of Schemes
105(5)
3.5 Procedure for Second-Order PDEs
110(9)
3.5.1 First Approach: Direct Variation
111(4)
3.5.2 Second Approach: System of PDEs
115(4)
3.6 Preliminaries on Discrete Functional Analysis
119(10)
3.6.1 Discrete Function Spaces
119(2)
3.6.2 Discrete Inequalities
121(5)
3.6.3 Discrete Gronwall Lemma
126(3)
4 Applications 129(98)
4.1 Target PDEs 1
129(26)
4.1.1 Cahn–Hilliard Equation
129(20)
4.1.2 Allen–Cahn Equation
149(4)
4.1.3 Fisher–Kolmogorov Equation
153(2)
4.2 Target PDEs 2
155(9)
4.2.1 Korteweg–de Vries Equation
157(2)
4.2.2 Zakharov–Kuznetsov Equation
159(5)
4.3 Target PDEs 3
164(3)
4.3.1 Complex-Valued Ginzburg–Landau Equation
164(1)
4.3.2 Newell–Whitehead Equation
165(2)
4.4 Target PDEs 4
167(15)
4.4.1 Nonlinear SchrOdinger Equation
167(13)
4.4.2 Gross–Pitaevskii Equation
180(2)
4.5 Target PDEs 5
182(3)
4.5.1 Zakharov Equations
183(2)
4.6 Target PDEs 7
185(6)
4.6.1 Nonlinear Klein–Gordon Equation
185(4)
4.6.2 Shimoji–Kawai Equation
189(2)
4.7 Other Equations
191(36)
4.7.1 Keller–Segel Equation
191(4)
4.7.2 Camassa–Holm Equation
195(17)
4.7.3 Benjamin–Bona–Mahony Equation
212(10)
4.7.4 Feng Equation
222(5)
5 Advanced Topic I: Design of High-Order Schemes 227(44)
5.1 Orders of Accuracy of Schemes
227(2)
5.2 Spatially High-Order Schemes
229(18)
5.2.1 Discrete Symbols and Formulas
229(2)
5.2.2 Discrete Variational Derivative
231(2)
5.2.3 Design of Schemes
233(5)
5.2.4 Application Examples
238(9)
5.3 Temporally High-Order Schemes: Composition Method
247(1)
5.4 Temporally High-Order Schemes: High-Order Discrete Variational Derivatives
248(23)
5.4.1 Discrete Symbols
249(1)
5.4.2 Central Idea for High-Order Discrete Derivative
250(1)
5.4.3 Temporally High-Order Discrete Variational Derivative and Design of Schemes
251(20)
6 Advanced Topic II: Design of Linearly Implicit Schemes 271(22)
6.1 Basic Idea for Constructing Linearly Implicit Schemes
271(3)
6.2 Multiple-Points Discrete Variational Derivative
274(3)
6.2.1 For Real-Valued PDEs
274(1)
6.2.2 For Complex-Valued PDEs
275(2)
6.3 Design of Schemes
277(3)
6.3.1 For Real-Valued PDEs
277(2)
6.3.2 For Complex-Valued PDEs
279(1)
6.4 Applications
280(8)
6.4.1 Cahn—Hilliard Equation
280(3)
6.4.2 Odd-Order Nonlinear Schrodinger Equation
283(1)
6.4.3 Ginzburg—Landau Equation
283(1)
6.4.4 Zakharov Equations
284(1)
6.4.5 Newell—Whitehead Equation
285(3)
6.5 Remarks on the Stability of Linearly Implicit Schemes
288(5)
7 Advanced Topic III: Further Remarks 293(60)
7.1 Solving System of Nonlinear Equations
293(5)
7.1.1 Use of Numerical Newton Method Libraries
294(1)
7.1.2 Variants of Newton Method
295(1)
7.1.3 Spectral Residual Methods
296(2)
7.1.4 Implementation as a Predictor—Corrector Method
298(1)
7.2 Switch to Galerkin Framework
298(50)
7.2.1 Design of Galerkin Schemes
299(10)
7.2.2 Application Examples
309(39)
7.3 Extension to Non-Rectangular Meshes on 2D Region
348(5)
Appendix A Semi-Discrete Schemes in Space 353(4)
Appendix B Proof of Proposition 3.4 357(2)
Bibliography 359(14)
Index 373
Daisuke Furihata, Takayasu Matsuo