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Discrete variational derivative method : a structure-preserving numerical method for partial differential equations

Author: Daisuke Furihata; Takayasu Matsuo
Publisher: Boca Raton, FL : Chapman and Hall/CRC, ©2011.
Series: Chapman & Hall/CRC numerical analysis and scientific computing.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"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  Read more...
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Document Type: Book
All Authors / Contributors: Daisuke Furihata; Takayasu Matsuo
ISBN: 9781420094459 1420094459
OCLC Number: 678923740
Description: xi, 376 pages : illustrations ; 25 cm.
Contents: 1. Introduction and summary of this book --
An introductory example: the spinodal decomposition --
History --
Derivation of dissipative or conservative schemes advanced topics --
2. Target partial differential equations --
Variational derivatives --
First-order real-valued PDEs --
First-order complex-valued PDEs --
Systems of first-order PDEs --
Second-order PDEs --
3. 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 --
Procedure for second-order PDEs --
Preliminaries on discrete functional analysis --
4. Applications --
Target PDEs 1 --
Target PDEs 2 --
Target PDEs 3 --
Target PDEs 4 --
Target PDEs 5 --
Target PDEs 7 --
Other equations --
5. Advanced topic I: design of high-order schemes --
Orders of accuracy of schemes --
Spatially high-order schemes --
Temporally high-order schemes: composition method --
Temporally high-order schemes: high-order discrete variational derivatives --
6. Advanced topic II: design of linearly-implicit schemes --
Basic idea for constructing linearly-implicit schemes --
Multiple-points discrete variational derivative --
Design of schemes --
Applications --
Remarks on the stability of linearly-implicit schemes --
7. Advanced topic III: further remarks --
Solving system of nonlinear equations --
Switch to Galerkin framework --
Extension to non-rectangular meshes on 2D Region --
A. Semi-discrete schemes in space B --
Proof of Proposition 3.4.
Series Title: Chapman & Hall/CRC numerical analysis and scientific computing.
Responsibility: Daisuke Furihata, Takayasu Matsuo.

Abstract:

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  Read more...

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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 Read more...

 
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