Implementing spectral methods for partial differential equations : algorithms for scientists and engineers / David A. Kopriva.

Format:

Book

Author/Creator:

Kopriva, David A.

Series:

Scientific computation

Language:

English

Subjects (All):

Differential equations, Partial.

Spectral theory (Mathematics).

Physical Description:

xviii, 394 pages : illustrations ; 24 cm.

Place of Publication:

[Dordrecht] : Springer, [2009]

Summary:

This book offers a systematic and self-contained approach to solve partial differential equations numerically using single and multidomain spectral methods. It contains detailed algorithms in pseudocode for the application of spectral approximations to both one and two dimensional PDEs of mathematical physics describing potentials, transport, and wave propagation. David Kopriva, a well-known researcher in the field with extensive practical experience, shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries. The book addresses computational and applications scientists, as it emphasizes the practical derivation and implementation of spectral methods over abstract mathematics. It is divided into two parts: First comes a primer on spectral approximation and the basic algorithms, including FFT algorithms, Gauss quadrature algorithms, and how to approximate derivatives. The second part shows how to use those algorithms to solve steady and time dependent PDEs in one and two space dimensions. Exercises and questions at the end of each chapter encourage the reader to experiment with the algorithms.

Contents:

Part I Approximating Functions, Derivatives and Integrals

1 Spectral Approximation 3

1.1 Preamble: Series Solution of PDEs 3

1.2 The Fourier Basis Functions and Fourier Series 4

1.3 Series Truncation 6

1.4 Modal vs. Nodal Approximation 11

1.5 Discrete Orthogonality and Quadrature 11

1.6 Fourier Interpolation 14

1.6.1 Direct Computation of the Fourier Interpolation 17

1.6.2 Error of the Fourier Interpolation 19

1.7 The Derivative of the Fourier Interpolant 21

1.8 Polynomial Basis Functions 23

1.8.1 The Legendre Polynomials 24

1.8.2 The Chebyshev Polynomials 25

1.9 Polynomial Series 26

1.10 Polynomial Series Truncation 28

1.10.1 Derivatives of Truncated Series 30

1.11 Polynomial Quadrature 31

1.12 Orthogonal Polynomial Interpolation 35

2 Algorithms for Periodic Functions 39

2.1 How to Compute the Discrete Fourier Transform 39

2.1.1 Fourier Transforms of Complex Sequences 40

2.1.2 Fourier Transforms of Real Sequences 43

2.1.3 The Fourier Transform in Two Space Variables 48

2.2 The Real Fourier Transform 50

2.3 How to Evaluate the Fourier Interpolation Derivative by FFT 53

2.4 How to Compute Derivatives by Matrix Multiplication 54

3 Algorithms for Non-Periodic Functions 59

3.1 How to Compute the Legendre and Chebyshev Polynomials 59

3.2 How to Compute the Gauss Quadrature Nodes and Weights 62

3.2.1 Legendre Gauss Quadrature 62

3.2.2 Legendre Gauss-Lobatto Quadrature 64

3.2.3 Chebyshev Gauss Quadratures 67

3.3 How to Evaluate Chebyshev Interpolants via the FFT 67

3.3.1 The Fast Cheyshev Transform 68

3.4 How to Evaluate Polynomial Interpolants in Lagrange Form 73

3.5 How to Evaluate Polynomial Derivatives 78

3.5.1 Direct Evaluation of the Derivative 79

3.5.2 Evaluation of Derivatives by Matrix Multiplication 81

3.5.3 Even-Odd Decomposition 82

3.5.4 Evaluation by Transform Methods 84

3.5.5 Performance of Various Polynomial Derivative Algorithms 84

Part II Approximating Solutions of PDEs

4 Survey of Spectral Approximations 91

4.1 The Fourier Collocation Method 94

4.1.1 How to Implement the Fourier Collocation Method 96

4.1.2 Benchmark Solution 99

4.2 The Fourier Galerkin Method 101

4.2.1 How to Implement the Fourier Galerkin Method 103

4.2.2 Benchmark Solution 106

4.3 Nonlinear and Product Terms 107

4.3.1 The Galerkin Approximation 107

4.3.2 How to Compute the Convolution Sum 109

4.3.3 The Collocation Approximation 112

4.4 Polynomial Collocation Methods 115

4.4.1 Approximation of the Diffusion Equation 115

4.4.2 How to Implement the Methods 117

4.4.3 Benchmark Solution 119

4.4.4 Approximation of Scalar Advection 120

4.5 The Legendre Galerkin Method 123

4.5.1 How to Implement the Method 127

4.6 The Nodal Continuous Galerkin Method 129

4.6.1 How to Implement the Method 133

4.6.2 Benchmark Solution 134

4.7 The Nodal Discontinuous Galerkin Method 134

4.7.1 How to Implement the Method 138

4.7.2 Benchmark Solution 143

4.8 Summary and Some Broad Generalizations 144

5 Spectral Approximation on the Square 149

5.1 Approximation of Functions in Multiple Space Dimensions 149

5.2 Potential Problems on the Square 151

5.2.1 The Collocation Approximation 152

5.2.2 The Nodal Galerkin Approximation 173

5.3 Approximation of Time Dependent Advection-Diffusion 188

5.3.1 The Collocation Approximation 188

5.3.2 The Nodal Galerkin Approximation 189

5.3.3 Time Integration 191

5.3.4 How to Implement the Approximations 193

5.3.5 Benchmark Solution: Advection and Diffusion of a Spot in a Uniform Flow 200

5.4 Approximation of Wave Propagation Problems 202

5.4.1 The Nodal Discontinuous Galerkin Approximation 204

5.4.2 How to Implement the Nodal Discontinuous Galerkin Approximation 212

5.4.3 Benchmark Solution: Plane Wave Propagation 216

5.4.4 Benchmark Solution: Propagation of a Circular Sound Wave 217

6 Transformation Methods from Square to Non-Square Geometries 223

6.1 Mappings and Coordinate Transformations 223

6.1.1 Mapping a Straight Sided Quadrilateral 224

6.1.2 How to Approximate Curved Boundaries 225

6.1.3 How to Map the Reference Square to a Curved-Sided Quadrilateral 229

6.2 Transformation of Equations under Mappings 231

6.2.1 Two-Dimensional Forms 238

6.3 How to Approximate the Metric Terms 240

6.4 How to Compute the Metric Terms 242

7 Spectral Methods in Non-Square Geometries 247

7.1 Steady Potentials in a Quadrilateral Domain 247

7.1.1 The Collocation Approximation 247

7.1.2 The Nodal Galerkin Approximation 252

7.1.3 Solution of the Linear Systems 254

7.1.4 Benchmark Solution: Potential in Non-Square Domains 259

7.1.5 Benchmark Solution: Incompressible Flow over a Circular Obstacle 261

7.2 Steady Potentials in an Annulus 264

7.2.1 Benchmark Solution: Potential in an Annulus with a Source 271

7.3 Advection and Diffusion in Quadrilateral Domains 272

7.3.1 Transformation of the Advection-Diffusion Equation 272

7.3.2 The Collocation Approximation 273

7.3.3 The Nodal Galerkin Approximation 274

7.3.4 How to Implement the Approximations 275

7.3.5 Benchmark Solution: Advection and Diffusion in a Non-Square Geometry 276

7.3.6 Benchmark Solution: Advection and Diffusion of a Pollutant in a Curved Channel 277

7.4 Conservation Laws in Quadrilateral Domains 279

7.4.1 The Nodal Discontinuous Galerkin Approximation 280

7.4.2 How to Implement the Nodal Discontinuous Galerkin Approximation 282

7.4.3 Benchmark Solution: Acoustic Scattering off a Cylinder 285

8 Spectral Element Methods 293

8.1 Spectral Element Methods in One Space Dimension 296

8.1.1 The Continuous Galerkin Spectral Element Method 297

8.1.2 How to Implement the Continuous Galerkin Spectral Element Method 301

8.1.3 Benchmark Solution: Cooling of a Temperature Spot 305

8.1.4 The Discontinuous Galerkin Spectral Element Method 308

8.1.5 How to Implement the Discontinuous Galerkin Spectral Element Method 310

8.1.6 Benchmark Solution: Wave Propagation and Reflection 315

8.2 The Two-Dimensional Mesh and Its Specification 317

8.2.1 How to Construct a Two-Dimensional Mesh 321

8.2.2 Benchmark Solution: A Spectral Element Mesh for a Disk 326

8.3 The Spectral Element Method in Two Space Dimensions 326

8.3.1 How to Implement the Spectral Element Method 331

8.3.2 Benchmark Solution: Steady Temperatures in a Long Cylindrical Rod 340

8.4 The Discontinuous Galerkin Spectral Element Method 341

8.4.1 How to Implement the Discontinuous Galerkin Spectral Element Method 343

8.4.2 Benchmark Solution: Propagation of a Circular Wave in a Circular Domain 344

8.4.3 Benchmark Solution: Transmission and Reflection from a Material Interface 347.

Notes:

Includes bibliographical references (page 385) and indexes.

ISBN:

9789048122608

9048122600

9048122619

9789048122615

OCLC:

310400971