Discrete Variational Derivative Method : A Structure-Preserving Numerical Method for Partial Differential Equations

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 Cover......Page 1 Series: Numerical Analysis and Scientific Computing......Page 3 Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 10 1.1 An Introductory Example: Spinodal Decomposition......Page 14 1.2 History......Page 23 1.3.1 Procedure for First-Order Real-Valued PDEs......Page 25 1.3.2 Procedure for First-Order Complex-Valued PDEs......Page 32 1.3.3 Procedure for Systems of First-Order PDEs......Page 37 1.3.4 Procedure for Second-Order PDEs......Page 40 1.4.1 Design of Higher-Order Schemes......Page 47 1.4.2 Design of Linearly Implicit Schemes......Page 53 1.4.3 Further Remarks......Page 60 2.1 Variational Derivatives......Page 62 2.2 First-Order Real-Valued PDEs......Page 65 2.3 First-Order Complex-Valued PDEs......Page 71 2.4 Systems of First-Order PDEs......Page 73 2.5 Second-Order PDEs......Page 78 3.1 Discrete Symbols and Formulas......Page 82 3.2.1 Discrete Variational Derivative: Real-Valued Case......Page 88 3.2.2 Design of Schemes......Page 93 3.2.3 User's Choices......Page 100 3.3.1 Discrete Variational Derivative: Complex-Valued Case......Page 106 3.3.2 Design of Schemes......Page 109 3.4 Procedure for Systems of First-Order PDEs......Page 114 3.4.1 Design of Schemes......Page 118 3.5 Procedure for Second-Order PDEs......Page 123 3.5.1 First Approach: Direct Variation......Page 124 3.5.2 Second Approach: System of PDEs......Page 128 3.6.1 Discrete Function Spaces......Page 132 3.6.2 Discrete Inequalities......Page 134 3.6.3 Discrete Gronwall Lemma......Page 139 4.1.1 Cahn–Hilliard Equation......Page 142 4.1.2 Allen–Cahn Equation......Page 162 4.1.3 Fisher–Kolmogorov Equation......Page 166 4.2 Target PDEs 2......Page 168 4.2.1 Korteweg–de Vries Equation......Page 170 4.2.2 Zakharov–Kuznetsov Equation......Page 172 4.3.1 Complex-Valued Ginzburg–Landau Equation......Page 177 4.3.2 Newell–Whitehead Equation......Page 178 4.4.1 Nonlinear Schrödinger Equation......Page 180 4.4.2 Gross–Pitaevskii Equation......Page 193 4.5 Target PDEs 5......Page 195 4.5.1 Zakharov Equations......Page 196 4.6.1 Nonlinear Klein–Gordon Equation......Page 198 4.6.2 Shimoji–Kawai Equation......Page 202 4.7.1 Keller–Segel Equation......Page 204 4.7.2 Camassa–Holm Equation......Page 208 4.7.3 Benjamin–Bona–Mahony Equation......Page 225 4.7.4 Feng Equation......Page 235 5.1 Orders of Accuracy of Schemes......Page 240 5.2.1 Discrete Symbols and Formulas......Page 242 5.2.2 Discrete Variational Derivative......Page 244 5.2.3 Design of Schemes......Page 246 5.2.4 Application Examples......Page 251 5.3 Temporally High-Order Schemes: Composition Method......Page 260 5.4 Temporally High-Order Schemes: High-Order Discrete Variational Derivatives......Page 261 5.4.1 Discrete Symbols......Page 262 5.4.2 Central Idea for High-Order Discrete Derivative......Page 263 5.4.3 Temporally High-Order Discrete Variational Derivative and Design of Schemes......Page 264 6.1 Basic Idea for Constructing Linearly Implicit Schemes......Page 284 6.2.1 For Real-Valued PDEs......Page 287 6.2.2 For Complex-Valued PDEs......Page 288 6.3.1 For Real-Valued PDEs......Page 290 6.3.2 For Complex-Valued PDEs......Page 292 6.4.1 Cahn–Hilliard Equation......Page 293 6.4.3 Ginzburg–Landau Equation......Page 296 6.4.4 Zakharov Equations......Page 297 6.4.5 Newell–Whitehead Equation......Page 298 6.5 Remarks on the Stability of Linearly Implicit Schemes......Page 301 7.1 Solving System of Nonlinear Equations......Page 306 7.1.1 Use of Numerical Newton Method Libraries......Page 307 7.1.2 Variants of Newton Method......Page 308 7.1.3 Spectral Residual Methods......Page 309 7.2 Switch to Galerkin Framework......Page 311 7.2.1 Design of Galerkin Schemes......Page 312 7.2.2 Application Examples......Page 322 7.3 Extension to Non-Rectangular Meshes on 2D Region......Page 361 Appendix A: Semi-Discrete Schemes in Space......Page 366 Appendix B: Proof of Proposition 3.4......Page 370 Bibliography......Page 372 "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