High Order Difference Methods for Time Dependent PDE

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Springer Science & Business Media, Dec 6, 2007 - Mathematics - 334 pages
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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.
 

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Contents

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Page 326 - ... A. (1994). Constitutive inconsistency: rigorous solution of Maxwell equations based on a dual approach. IEEE Trans. Magn. 30, 3586-3589. Guibas, L., and Stolfi, J. (1985). Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Trans. Graphics 4, 74-123. Gustafsson, B., Kreiss, H.-O., and Oliger, J. (1995). Time Dependent Problems and Difference Methods. New York: Wiley. Hocking, JG, and Young, GS (1988). Topology. New York: Dover. Hurewicz, W., and...
Page 326 - E. (1981). Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II. Math. Comp., 36:605-626.
Page 326 - Gustafsson (1998). On the implementation of boundary conditions for the method of lines BIT 38, pp 293-314.

About the author (2007)

Bertil Gustafsson is a professor with the Department of Scientific Computing at Uppsala University, Sweden.

Heinz-Otto Kreiss is a professor with the UCLA Department of Mathematics. He is the coauthor of Initial-Boundary Value Problems and the Navier-Stokes Equations.

Joseph Oliger is a professor with the Department of Computer Science at Stanford University.

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