Discrete Variational Derivative Method: A Structure-Preserving Numerical Method

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Discrete Variational Derivative Method by Daisuke Furihata, Takayasu Matsuo

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 introduces key factors and basic ideas of discrete variational derivative method.

FORMAT
Hardcover LANGUAGE
English CONDITION
Brand New Publisher 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 Schrodinger equation (optical waves) "Dissipative" equations such as the Cahn--Hilliard equation (some phase separation phenomena) and the Newell-Whitehead equation (two-dimensional Benard 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

Back Cover

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.

Table of Contents

Introduction and Summary of This Book. Target Partial Differential Equations. Discrete Variational Derivative Method. Applications. Advanced Topic I: Design of High-Order Schemes. Advanced Topic II: Design of Linearly-Implicit Schemes. Advanced Topic III: Further Remarks.

Review

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

Review Quote

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. --R

Details ISBN1420094459 Author Takayasu Matsuo Short Title DISCRETE VARIATIONAL DERIVATIV Language English ISBN-10 1420094459 ISBN-13 9781420094459 Media Book Format Hardcover Imprint Chapman & Hall/CRC DEWEY 515.355 Pages 376 Series Chapman & Hall/CRC Numerical Analysis and Scientific Computing Series Illustrations 12 Tables, black and white; 100 Illustrations, black and white Affiliation University of Tokyo, Bunkyo, Japan Subtitle A Structure-Preserving Numerical Method for Partial Differential Equations DOI 10.1604/9781420094459 AU Release Date 2010-12-09 NZ Release Date 2010-12-09 US Release Date 2010-12-09 Publisher Taylor & Francis Ltd Year 2010 Publication Date 2010-12-09 Audience Professional & Vocational Country of Publication United Kingdom Country of Origin GB Product Class Description Computing: General UK Release Date 2010-12-14
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