1st Edition
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
Preface
Introduction and Summary of This Book
An Introductory Example: the Spinodal Decomposition
History
Derivation of Dissipative or Conservative Schemes
Advanced Topics
Target Partial Differential Equations
Variational Derivatives
First-Order Real-Valued PDEs
First-Order Complex-Valued PDEs
Systems of First-Order PDEs
Second-Order PDEs
Discrete Variational Derivative Method
Discrete Symbols and Formulas
Procedure for First-Order Real-Valued PDEs
Procedure for First-Order Complex-Valued PDEs
Procedure for Systems of First-Order PDEs
Design of Schemes
Procedure for Second-Order PDEs
Preliminaries on Discrete Functional Analysis
Applications
Target PDEs
Cahn–Hilliard Equation
Allen–Cahn Equation
Fisher–Kolmogorov Equation
Target PDEs
Target PDEs
Target PDEs
Nonlinear Schr¨odinger Equation
Target PDEs
Zakharov Equations
Target PDEs
Other Equations
Advanced Topic I: Design of High-Order Schemes
Orders of Accuracy of the Schemes
Spatially High-Order Schemes
Temporally High-Order Schemes: With the Composition Method
Temporally High-Order Schemes: With High-Order Discrete Variational Derivatives
Advanced Topic II: Design of Linearly-Implicit Schemes
Basic Idea for Constructing Linearly-Implicit Schemes
Multiple-Points Discrete Variational Derivative
Design of Schemes
Applications
Remark on the Stability of Linearly-Implicit Schemes
Advanced Topic III: Further Remarks
Solving System of Nonlinear Equations
Switch to Galerkin Framework
Extension to Non-Rectangular Meshes on D Region
A Semi-discrete schemes in space
B Proof of Proposition 3.4
Bibliography
Index
Biography
Daisuke Furihata, Takayasu Matsuo
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
