High order difference methods for time dependent PDE / Bertil Gustafsson.

Format:

Book

Author/Creator:

Gustafsson, Bertil, 1930-

Contributor:

Louis A. Duhring Fund.

Series:

Springer series in computational mathematics 0179-3632 ; 38.

Springer series in computational mathematics, 0179-3632 ; 38

Language:

English

Subjects (All):

Differential equations, Partial--Numerical solutions.

Differential equations, Partial.

Finite differences.

Physical Description:

xiii, 334 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2008]

Summary:

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.

Contents:

1 When are High Order Methods Effective? 1

1.2 Wave Propagation Problems 2

1.3 Parabolic Equations 8

1.4 Schrodinger Type Equations 11

2 Well-posedness and Stability 13

2.1 Well Posed Problems 13

2.2 Periodic Problems and Fourier Analysis 16

2.2.1 The PDE Problem 17

2.2.2 Difference Approximations 21

2.3 Initial-Boundary Value Problems and the Energy Method 29

2.3.1 The PDE Problem 29

2.3.2 Semidiscrete Approximations 33

2.3.3 Fully Discrete Approximations 38

2.4 Initial-Boundary Value Problems and Normal Mode Analysis for Hyperbolic Systems 41

2.4.1 Semidiscrete Approximations 41

2.4.2 Fully Discrete Approximations 59

3 Order of Accuracy and the Convergence Rate 69

3.1 Periodic Solutions 69

3.2 Initial-Boundary Value Problems 72

4 Approximation in Space 81

4.1 High Order Formulas on Standard Grids 81

4.2 High Order Formulas on Staggered Grids 85

4.3 Compact Pade Type Difference Operators 87

4.4 Optimized Difference Operators 91

5 Approximation in Time 95

5.1 Stability and the Test Equation 95

5.2 Runge-Kutta Methods 97

5.3 Linear Multistep Methods 102

5.4 Deferred Correction 108

5.5 Richardson Extrapolation 111

6 Coupled Space-Time Approximations 115

6.1 Taylor Expansions and the Lax-Wendroff Principle 115

6.2 Implicit Fourth Order Methods 117

7 Boundary Treatment 127

7.1 Numerical Boundary Conditions 127

7.2 Summation by Parts (SBP) Difference Operators 130

7.3 SBP Operators and Projection Methods 140

7.4 SBP Operators and Simultaneous Approximation Term (SAT) Methods 147

8 The Box Scheme 157

8.1 The Original Box Scheme 157

8.2 The Shifted Box Scheme 161

8.3 Two Space Dimensions 165

8.4 Nonuniform Grids 169

9 Wave Propagation 177

9.1 The Wave Equation 177

9.1.1 One Space Dimension 178

9.1.2 Two Space Dimensions 185

9.2 Discontinuous Coefficients 192

9.2.1 The Original One Step Scheme 193

9.2.2 Modified Coefficients 201

9.2.3 An Example with Discontinuous Solution 206

9.3 Boundary Treatment 209

9.3.1 High Order Boundary Conditions 209

9.3.2 Embedded Boundaries 210

10 A Problem in Fluid Dynamics 219

10.1 Large Scale Fluid Problems and Turbulent Flow 219

10.2 Stokes Equations for Incompressible Flow 220

10.3 A Fourth Order Method for Stokes Equations 223

10.4 Navier-Stokes Equations for Incompressible Flow 228

10.5 A Fourth Order Method for Navier-Stokes Equations 231

11 Nonlinear Problems with Shocks 245

11.1 Difference Methods and Nonlinear Equations 245

11.2 Conservation Laws 246

11.3 Shock Fitting 251

11.4 Artificial Viscosity 252

11.5 Upwind Methods 257

11.6 ENO and WENO Schemes 261

12 Introduction to Other Numerical Methods 267

12.1 Finite Element Methods 267

12.1.1 Galerkin FEM 267

12.1.2 Petrov-Galerkin FEM 281

12.2 Discontinuous Galerkin Methods 283

12.3 Spectral Methods 289

12.3.1 Fourier Methods 290

12.3.2 Polynomial Methods 295

12.4 Finite Volume Methods 300

A Solution of Difference Equations 307

B The Form of SBP Operators 311

B.1 Diagonal H-norm 311

B.2 Full H[subscript 0]-norm 317

B.3 A Pade Type Operator 323.

Notes:

Includes bibliographical references (pages 325-330).

Local Notes:

Acquired for the Penn Libraries with assistance from the Louis A. Duhring Fund.

ISBN:

9783540749929

3540749926

OCLC:

181090541