## High Order Difference Methods for Time Dependent PDEMany 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

XIV | 41 |

XVI | 66 |

XVII | 69 |

XX | 72 |

XXI | 79 |

XXII | 81 |

XXIV | 85 |

XXV | 87 |

XXVI | 91 |

XXVII | 93 |

XXVIII | 95 |

XXX | 97 |

XXXI | 102 |

XXXII | 108 |

XXXIII | 111 |

XXXIV | 113 |

XXXV | 115 |

XXXVIII | 124 |

XXXIX | 127 |

XLII | 130 |

XLIII | 140 |

XLIV | 147 |

XLV | 155 |

XLVI | 157 |

XLIX | 161 |

L | 176 |

LI | 177 |

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### Common terms and phrases

4th order applied assume basis functions boundary conditions box scheme Chapter characteristic equation coefficients computational conservation law consider constant construct defined derivatives diagonal difference approximation difference operator difference scheme differential equation differential operator discontinuous discrete eigenvalues energy method example finite volume methods formulas Fourier transform fourth order Galerkin Galerkin methods grid function grid points Gustafsson high order higher order methods hyperbolic imaginary axis initial data initial–boundary value problem Kreiss condition L2-norm Laplace transform linear matrix norm normal mode analysis notation numerical solution obtained ODE system order accuracy order approximation order of accuracy order scheme PadŽe piecewise polynomials Re˜s Runge-Kutta methods SBP operators scalar product second order second order approximation Section semibounded semidiscrete shock space dimensions stability domain step scheme Taylor expansion Theorem truncation error u(xt un+1 un+1j vector zero κμ

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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.