Implementing spectral methods for partial differential equations: algorithms for scientists and engineers
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
2009
|
| Series: | Scientific computation
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XVIII, 394 S. Ill., graph. Darst. |
| ISBN: | 9789048122615 9789048122608 |
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MARC
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Record in the Search Index
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| adam_text | Scientific Computation
D.A.
Kopriva
Implementing Spectral Methods for Partial Differential Equations
Algorithms for Scientists and Engineers
This book otters 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 tew 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 stead) 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
Preface
.....................................
vii
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 Chebyshev Transform
............... 68
3.4
How to Evaluate Polynomial
Interpolants
in
Lagrange Form .... 73
ix
x
Contents
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
Contents
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
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
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
Spectral Element Methods
........................ 293
8.1
Spectral Element Methods in One Space Dimension
........ 296
8.1.1
The Continuous Galerkin Spectral Element Method
.... 297
xii Contents
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
A Pseudocode Conventions
.........................355
В
Floating Point Arithmetic
........................359
С
Basic Linear Algebra Subroutines
(BLAS)...............361
D
Linear Solvers
...............................363
D.I Direct Solvers
............................363
D.I.I Tri-Diagonal Solver
.....................363
D.1.2
LU
Factorization
.......................364
D.2 Iterative Solvers
...........................368
E
Data Structures
..............................373
E.I Linked Lists
.............................373
E.I.I Example: Elements that Share a Node
............376
E.2 Hash Tables
.............................377
E.2.1 Example: Avoiding Duplicate Edges in a Mesh
.......381
References
...................................385
Index of Algorithms
..............................387
Subject Index
..................................389
|
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| building | Verbundindex |
| bvnumber | BV035768777 |
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| ctrlnum | (OCoLC)310400971 (DE-599)DNB992368359 |
| dewey-full | 515.7222 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.7222 |
| dewey-search | 515.7222 |
| dewey-sort | 3515.7222 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035768777 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:04:06Z |
| institution | BVB |
| isbn | 9789048122615 9789048122608 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018628517 |
| oclc_num | 310400971 |
| open_access_boolean | |
| owner | DE-703 DE-11 DE-20 DE-91G DE-BY-TUM DE-634 DE-83 |
| owner_facet | DE-703 DE-11 DE-20 DE-91G DE-BY-TUM DE-634 DE-83 |
| physical | XVIII, 394 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series2 | Scientific computation |
| spelling | Kopriva, David A. Verfasser (DE-588)140046666 aut Implementing spectral methods for partial differential equations algorithms for scientists and engineers David A. Kopriva Berlin Springer 2009 XVIII, 394 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Scientific computation Differential equations, Partial Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Lehrbuch (DE-588)4123623-3 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Spektralmethode (DE-588)4224817-6 s Numerische Mathematik (DE-588)4042805-9 s Algorithmus (DE-588)4001183-5 s Lehrbuch (DE-588)4123623-3 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628517&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628517&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Kopriva, David A. Implementing spectral methods for partial differential equations algorithms for scientists and engineers Differential equations, Partial Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Lehrbuch (DE-588)4123623-3 gnd |
| subject_GND | (DE-588)4224817-6 (DE-588)4044779-0 (DE-588)4001183-5 (DE-588)4042805-9 (DE-588)4123623-3 |
| title | Implementing spectral methods for partial differential equations algorithms for scientists and engineers |
| title_auth | Implementing spectral methods for partial differential equations algorithms for scientists and engineers |
| title_exact_search | Implementing spectral methods for partial differential equations algorithms for scientists and engineers |
| title_full | Implementing spectral methods for partial differential equations algorithms for scientists and engineers David A. Kopriva |
| title_fullStr | Implementing spectral methods for partial differential equations algorithms for scientists and engineers David A. Kopriva |
| title_full_unstemmed | Implementing spectral methods for partial differential equations algorithms for scientists and engineers David A. Kopriva |
| title_short | Implementing spectral methods for partial differential equations |
| title_sort | implementing spectral methods for partial differential equations algorithms for scientists and engineers |
| title_sub | algorithms for scientists and engineers |
| topic | Differential equations, Partial Spectral theory (Mathematics) Spektralmethode (DE-588)4224817-6 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Algorithmus (DE-588)4001183-5 gnd Numerische Mathematik (DE-588)4042805-9 gnd Lehrbuch (DE-588)4123623-3 gnd |
| topic_facet | Differential equations, Partial Spectral theory (Mathematics) Spektralmethode Partielle Differentialgleichung Algorithmus Numerische Mathematik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628517&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018628517&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT koprivadavida implementingspectralmethodsforpartialdifferentialequationsalgorithmsforscientistsandengineers |