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High Order Difference Methods for Time Dependent PDE

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ISBN-10: 3540749926

ISBN-13: 9783540749929

Edition: 2008

Authors: Bertil Gustafsson

List price: $149.99
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The subject of this book is high order finite difference methods for time dependent PDE. The idea is to give an overview of the basic theory and construction principles by using model examples. The book also contains a general presentation of the techniques and results for well-posedness and stability, with inclusion of the three fundamental methods of analysis both for PDE in its original and discretized form: the Fourier transform, the eneregy method and the Laplace transform. Various types of wave propagation problems are treated in specific detail since high order methods are particularly effective for these problems.
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Book details

List price: $149.99
Copyright year: 2008
Publisher: Springer Berlin / Heidelberg
Publication date: 12/7/2007
Binding: Hardcover
Pages: 334
Size: 6.10" wide x 9.25" long x 0.38" tall
Weight: 1.518
Language: English

When are High Order Methods Effective?
Wave Propagation Problems
Parabolic Equations
Schrodinger Type Equations
Well-posedness and Stability
Well Posed Problems
Periodic Problems and Fourier Analysis
The PDE Problem
Difference Approximations
Initial-Boundary Value Problems and the Energy Method
The PDE Problem
Semidiscrete Approximations
Fully Discrete Approximations
Initial-Boundary Value Problems and Normal Mode Analysis for Hyperbolic Systems
Semidiscrete Approximations
Fully Discrete Approximations
Order of Accuracy and the Convergence Rate
Periodic Solutions
Initial-Boundary Value Problems
Approximation in Space
High Order Formulas on Standard Grids
High Order Formulas on Staggered Grids
Compact Pade Type Difference Operators
Optimized Difference Operators
Approximation in Time
Stability and the Test Equation
Runge-Kutta Methods
Linear Multistep Methods
Deferred Correction
Richardson Extrapolation
Coupled Space-Time Approximations
Taylor Expansions and the Lax-Wendroff Principle
Implicit Fourth Order Methods
Boundary Treatment
Numerical Boundary Conditions
Summation by Parts (SBP) Difference Operators
SBP Operators and Projection Methods
SBP Operators and Simultaneous Approximation Term (SAT) Methods
The Box Scheme
The Original Box Scheme
The Shifted Box Scheme
Two Space Dimensions
Nonuniform Grids
Wave Propagation
The Wave Equation
One Space Dimension
Two Space Dimensions
Discontinuous Coefficients
The Original One Step Scheme
Modified Coefficients
An Example with Discontinuous Solution
Boundary Treatment
High Order Boundary Conditions
Embedded Boundaries
A Problem in Fluid Dynamics
Large Scale Fluid Problems and Turbulent Flow
Stokes Equations for Incompressible Flow
A Fourth Order Method for Stokes Equations
Navier-Stokes Equations for Incompressible Flow
A Fourth Order Method for Navier-Stokes Equations
Nonlinear Problems with Shocks
Difference Methods and Nonlinear Equations
Conservation Laws
Shock Fitting
Artificial Viscosity
Upwind Methods
ENO and WENO Schemes
Introduction to Other Numerical Methods
Finite Element Methods
Galerkin FEM
Petrov-Galerkin FEM
Discontinuous Galerkin Methods
Spectral Methods
Fourier Methods
Polynomial Methods
Finite Volume Methods
Solution of Difference Equations
The Form of SBP Operators
Diagonal H-norm
Full H[subscript 0]-norm
A Pade Type Operator