High Order Difference Methods for Time Dependent PDE

1. Front Matter

   Pages i-xv

   PDF

2. When are High Order Methods Effective?

   Pages 1-12

3. Well-posedness and Stability

   Pages 13-68

4. Order of Accuracy and the Convergence Rate

   Pages 69-80

5. Approximation in Space

   Pages 81-94

6. Approximation in Time

   Pages 95-114

7. Coupled Space-Time Approximations

   Pages 115-125

8. Boundary Treatment

   Pages 127-156

9. The Box Scheme

   Pages 157-176

10. Wave Propagation

    Pages 177-218

11. A Problem in Fluid Dynamics

    Pages 219-243

12. Nonlinear Problems with Shocks

    Pages 245-266

13. Introduction to Other Numerical Methods

    Pages 267-305

14. Back Matter

    Pages 307-334

    PDF

Many books have been written on ?nite difference methods (FDM), but there are good reasons to write still another one. The main reason is that even if higher order methods have been known for a long time, the analysis of stability, accuracy and effectiveness is missing to a large extent. For example, the de?nition of the formal high order accuracy is based on the assumption that the true solution is smooth, or expressed differently, that the grid is ?ne enough such that all variations in the solution are well resolved. In many applications, this assumption is not ful?lled, and then it is interesting to know if a high order method is still effective. Another problem that needs thorough analysis is the construction of boundary conditions such that both accuracy and stability is upheld. And ?nally, there has been quite a strongdevelopmentduringthe last years, inparticularwhenit comesto verygeneral and stable difference operators for application on initial–boundary value problems. The content of the book is not purely theoretical, neither is it a set of recipes for varioustypesof applications. The idea is to give an overviewof the basic theoryand constructionprinciplesfor differencemethodswithoutgoing into all details. For - ample, certain theorems are presented, but the proofs are in most cases left out. The explanation and application of the theory is illustrated by using simple model - amples.

• Approximation
• difference methods
• high order
• initial-boundary value problems
• partial differential equation
• stability
• wave propagation
• partial differential equations

From the reviews:

"This book presents the theory and construction principles of high order finite difference methods (FDM) for numerical solving of time dependent partial differential equations. … Many types of finite difference schemes are completely studied and numerical experiments and graphs are presented. … The book is written in a clear and comprehensive manner. It is recommended to researchers, PD students and readers interested in effective methods for numerical solving of partial differential equations." (Snezhana Gocheva-Ilieva, Zentralblatt MATH, Vol. 1146, 2008)

• Uppsala, Sweden

  Bertil Gustafsson

Policies and ethics