Bertil Gustafsson (auth.)3540749926, 978-3-540-74992-9
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. |
Table of contents :
Front Matter….Pages i-xv
When are High Order Methods Effective?….Pages 1-12
Well-posedness and Stability….Pages 13-68
Order of Accuracy and the Convergence Rate….Pages 69-80
Approximation in Space….Pages 81-94
Approximation in Time….Pages 95-114
Coupled Space-Time Approximations….Pages 115-125
Boundary Treatment….Pages 127-156
The Box Scheme….Pages 157-176
Wave Propagation….Pages 177-218
A Problem in Fluid Dynamics….Pages 219-243
Nonlinear Problems with Shocks….Pages 245-266
Introduction to Other Numerical Methods….Pages 267-305
Back Matter….Pages 307-334 |
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