Spis treści: Implementing Spectral Methods for Partial Differential Equations


Part I: Approximating Functions, Derivatives and Integrals

1. Spectral Approximation, s. 3

1.1     Preamble: Series Solution of PDEs, s. 3
1.2    The Fourier Basis Funetions and Fourier Series, s. 4
1.3    Series Truncation, s. 6
1.4    Modal vs. Nodal Approximation, s. 11
1.5     Discrete Orthogonality and Quadrature, s. 11
1.6    Fourier Interpolation, s. 14
1.6.1     Direct Computation of the Fourier Interpolation, s. 17
1.6.2     Error of the Fourier Interpolation, s. 19
1.7    The Derivative of the Fourier Interpolant, s. 21
1.8    Polynomial Basis Functions, s. 23
1.8.1     The Legendre Polynomials, s. 24
1.8.2     The Chebyshev Polynomials, s. 25
1.9    Polynomial Series, s. 26
1.10  Polynomial Series Truncation, s. 28
1.10.1   Derivatives of Truncated Series, s. 30
1.11   Polynomial Quadrature, s. 31
1.12  Orthogonal Polynomial Interpolation, s. 35

2. Algorithms for Periodic Functions, s. 39

2.1     How to Compute the Discrete Fourier Transform, s. 39
2.1.1     Fourier Transforms of Complex Sequences, s. 40
2.1.2     Fourier Transforms of Real Sequences, s. 43
2.1.3     The Fourier Transform in Two Space Variables, s. 48
2.2    The Real Fourier Transform, s. 50
2.3    How to Evaluate the Fourier Interpolation Derivative by FFT, s. 53
2.4    How to Compute Derivatives by Matrix Multiplication, s. 54

3. Algorithms for Non-Periodic Functions, s. 59

3.1     How to Compute the Legendre and Chebyshev Polynomials, s. 59
3.2    How to Compute the Gauss Quadrature Nodes and Weights, s. 62
3.2.1     Legendre Gauss Quadrature, s. 62
3.2.2     Legendre Gauss-Lobatto Quadrature, s. 64
3.2.3    Chebyshev Gauss Quadratures, s. 67
3.3     How to Evaluate Chebyshev Interpolants via the FFT, s. 67
3.3.1     The Fast Chebyshev Transform, s. 68
3.4    How to Evaluate Polynomial Interpolants in Lagrange Form, s. 73
3.5    How to Evaluate Polynomial Derivatives, s. 78
3.5.1     Direct Evaluation of the Derivative, s. 79
3.5.2     Evaluation of Derivatives by Matrix Multiplication, s. 81
3.5.3     Even-Odd Decomposition, s. 82
3.5.4     Evaluation by Transform Methods, s. 84
3.5.5     Performance of Various Polynomial Derivative Algorithms, s. 84

Part II: Approximating Solutions of PDEs

4      Survey of Spectral Approximations, s. 91
4.1     The Fourier Collocation Method, s. 94
4.1.1     How to Implement the Fourier Collocation Method, s. 96
4.1.2     Benchmark Solution, s. 99
4.2    The Fourier Galerkin Method, s. 101
4.2.1     How to Implement the Fourier Galerkin Method, s. 103
4.2.2     Benchmark Solution, s. 106
4.3    Nonlinear and Product Terms, s. 107
4.3.1     The Galerkin Approximation, s. 107
4.3.2    How to Compute the Convolution Sum, s. 109
4.3.3     The Collocation Approximation, s. 112
4.4    Polynomial Collocation Methods, s. 115
4.4.1     Approximation of the Diffusion Equation, s. 115
4.4.2     How to Implement the Methods, s. 117
4.4.3     Benchmark Solution, s. 119
4.4.4    Approximation of Scalar Advection, s. 120
4.5    The Legendre Galerkin Method, s. 123
4.5.1    How to Implement the Method, s. 127
4.6    The Nodal Continuous Galerkin Method, s. 129
4.6.1     How to Implement the Method, s. 133
4.6.2    Benchmark Solution, s. 134
4.7    The Nodal Discontinuous Galerkin Method, s. 134
4.7.1     How to Implement the Method, s. 138
4.7.2    Benchmark Solution, s. 143
4.8    Summary and Some Broad Generalizations, s. 144

5. Spectral Approximation on the Square, s. 149

5.1     Approximation of Functions in Multiple Space Dimensions, s. 149
5.2    Potential Problems on the Square, s. 151
5.2.1     The Collocation Approximation, s. 152
5.2.2     The Nodal Galerkin Approximation, s. 173
5.3    Approximation of Time Dependent Advection-Diffusio, s. 188
5.3.1     The Collocation Approximation, s. 188
5.3.2     The Nodal Galerkin Approximation, s. 189
5.3.3    Time Integration, s. 191
5.3.4     How to Implement the Approximations, s. 193
5.3.5     Benchmark Solution: Advection and Diffusion of a Spot in a Uniform Flow, s. 200
5.4    Approximation of Wave Propagation Problems, s. 202
5.4.1     The Nodal Discontinuous Galerkin Approximation, s. 204
5.4.2     How to Implement the Nodal Discontinuous Galerkin Approximation, s. 212
5.4.3     Benchmark Solution: Plane Wave Propagation, s. 216
5.4.4    Benchmark Solution: Propagation of a Circular Sound Wave, s. 217

6. Transformation Methods from Square to Non-Square Geometries, s. 223

6.1     Mappings and Coordinate Transformations, s. 223
6.1.1     Mapping a Straight Sided Quadrilateral, s. 224
6.1.2     How to Approximate Curved Boundaries, s. 225
6.1.3     How to Map the Reference Square to a Curved-Sided Quadrilateral, s. 229
6.2    Transformation of Equations under Mappings, s. 231
6.2.1    Two-Dimensional Forms, s. 238
6.3    How to Approximate the Metric Terms, s. 240
6.4    How to Compute the Metric Terms, s. 242

7. Spectral Methods in Non-Square Geometries, s. 247

7.1     Steady Potentials in a Quadrilateral Domain, s. 247
7.1.1     The Collocation Approximation, s. 247
7.1.2    The Nodal Galerkin Approximation, s. 252
7.1.3     Solution of the Linear Systems, s. 254
7.1.4    Benchmark Solution: Potential in Non-Square Domains, s. 259
7.1.5     Benchmark Solution: Incompressible Flow over a Circular Obstacle, s. 261
7.2    Steady Potentials in an Annulus, s. 264
7.2.1     Benchmark Solution: Potential in an Annulus with a Source, s. 271
7.3    Advection and Diffusion in Quadrilateral Domains, s. 272
7.3.1     Transformation of the Advection-Diffusion Equation, s. 272
7.3.2    The Collocation Approximation., s. 273
7.3.3    The Nodal Galerkin Approximation, s. 274
7.3.4     How to Implement the Approximations, s. 275
7.3.5     Benchmark Solution: Advection and Diffusion in a Non-Square Geometry, s. 276
7.3.6     Benchmark Solution: Advection and Diffusion of a Pollutant in a Curved Channel, s. 277
7.4    Conservation Laws in Quadrilateral Domains, s. 279
7.4.1     The Nodal Discontinuous Galerkin Approximation, s. 280
7.4.2    How to Implement the Nodal Discontinuous Galerkin Approximation, s. 282
7.4.3     Benchmark Solution: Acoustic Scattering off a Cylinder, s. 285

8. Spectral Element Methods, s. 293

8.1    Spectral Element Methods in One Space Dimension, s. 296
8.1.1    The Continuous Galerkin Spectral Element Method, s. 297
8.1.2     How to Implement the Continuous Galerkin Spectral Element Method, s. 301
8.1.3     Benchmark Solution: Cooling of a Temperature Spot, s. 305
8.1.4     The Discontinuous Galerkin Spectral Element Method, s. 308
8.1.5     How to Implement the Discontinuous Galerkin Spectral Element Method, s. 310
8.1.6     Benchmark Solution: Wave Propagation and Reflection, s. 315
8.2    The Two-Dimensional Mesh and Its Specification, s. 317
8.2.1     How to Construct a Two-Dimensional Mesh, s. 321
8.2.2     Benchmark Solution: A Spectral Element Mesh for a Disk, s. 326
8.3    The Spectral Element Method in Two Space Dimensions, s. 326
8.3.1     How to Implement the Spectral Element Method, s. 331
8.3.2     Benchmark Solution: Steady Temperatures in a Long Cylindrical Rod, s. 340
8.4    The Discontinuous Galerkin Spectral Element Method, s. 341
8.4.1     How to Implement the Discontinuous Galerkin Spectral Element Method, s. 343
8.4.2     Benchmark Solution: Propagation of a Circular Wave in a Circular Domain, s. 344
8.4.3     Benchmark Solution: Transmission and Reflection from a Material Interface, s. 347

A. Pseudocode Conventions, s. 355

B. Floating Point Arithmetic, s. 359

C. Basic Linear Algebra Subroutines (BLAS), s. 361

D. Linear Solvers, s. 363

D.1   Direct Solvers, s. 363
D. 1.1    Tri-Diagonal Solver, s. 363
D. 1.2   LU Factorization, s. 364
D.2   Iterative Solvers, s. 368

E. Data Structure, s. 373

E.1    Linked Lists, s. 373
E. 1.1    Example: Elements that Share a Node, s. 376
E.2   Hash Tables, s. 377
E.2.1    Example: Avoiding Duplicate Edges in a Mesh, s. 381

References, s. 385
Index of Algorithms, s. 387
Subject Index, s. 389

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