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

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