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

1st Edition

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

By Daisuke Furihata, Takayasu Matsuo Copyright 2011
376 Pages 100 B/W Illustrations
by Chapman & Hall

376 Pages
by Chapman & Hall
Also available as eBook on:
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... Read more

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

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