Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, 1st Edition (Hardback) book cover

Discrete Variational Derivative Method

A Structure-Preserving Numerical Method for Partial Differential Equations, 1st Edition

By Daisuke Furihata, Takayasu Matsuo

Chapman and Hall/CRC

376 pages | 100 B/W Illus.

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Description

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

Reviews

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

Table of Contents

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

About the Series

Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT021000
MATHEMATICS / Number Systems