Numerical methods for evolutionary differential equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia, Pa.
Soc. for Industrial and Applied Math.
2008
|
| Schriftenreihe: | Computational science & engineering
5 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 375 - 386 |
| Beschreibung: | XIII, 395 S. Ill., graph. Darst. |
| ISBN: | 9780898716528 |
Internformat
MARC
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| 020 | |a 9780898716528 |9 978-0-898716-52-8 | ||
| 035 | |a (OCoLC)192109753 | ||
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| 100 | 1 | |a Ascher, Uri M. |d 1946- |e Verfasser |0 (DE-588)136140823 |4 aut | |
| 245 | 1 | 0 | |a Numerical methods for evolutionary differential equations |c Uri M. Ascher |
| 264 | 1 | |a Philadelphia, Pa. |b Soc. for Industrial and Applied Math. |c 2008 | |
| 300 | |a XIII, 395 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Computational science & engineering |v 5 | |
| 500 | |a Literaturverz. S. 375 - 386 | ||
| 650 | 7 | |a Análise numérica |2 larpcal | |
| 650 | 7 | |a Equações diferenciais parciais |2 larpcal | |
| 650 | 4 | |a Équations d'évolution - Solutions numériques | |
| 650 | 4 | |a Evolution equations |x Numerical solutions | |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differentialgleichung |0 (DE-588)4012249-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Zeitabhängigkeit |0 (DE-588)4320088-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differentialgleichung |0 (DE-588)4012249-9 |D s |
| 689 | 0 | 1 | |a Zeitabhängigkeit |0 (DE-588)4320088-6 |D s |
| 689 | 0 | 2 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a Computational science & engineering |v 5 |w (DE-604)BV022382702 |9 5 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016571494 | ||
Datensatz im Suchindex
| _version_ | 1804137764467245056 |
|---|---|
| adam_text | Contents
Preface
χι
Introduction
1
1.1
Well-Posed
Initial Value
Problems..................... 4
1.1.1
Simple
model
cases
......................... 7
1.1.2
More general
cases
......................... 10
1.1.3
Initial
-boundary
value
problems
.................. 11
1.1.4
The solution
operator
........................ 12
1.2
A
Taste of Finite Differences........................
12
1.2.1
Stability ideas
............................ 17
1.3
Reviews
.................................. 24
1.3.1
Taylor s theorem
.......................... 24
1.3.2
Matrix norms and eigenvalues
................... 25
1.3.3
Function spaces
........................... 28
.3.4
The continuous Fourier transform
................. 29
.3.5
The matrix power and exponential
................. 30
.3.6
Fourier transform for periodic functions
.............. 31
1.4
Exercises
.................................. 32
Methods and Concepts for ODEs
37
2.1
Linear Multistep Methods
......................... 39
2.2
Runge-Kutta Methods
........................... 42
2.3
Convergence and O-stability
........................ 48
2.4
Error Control and Estimation
....................... 52
2.5
Stability of ODE Methods
......................... 53
2.6
Stiffness
.................................. 55
2.7
Solving Equations for Implicit Methods
.................. 59
2.8
Differential-Algebraic Equations
..................... 64
2.9
Symmetric and One-Sided Methods
.................... 66
2.10
Highly Oscillatory Problems
........................ 66
2.11
Boundary Value ODEs
........................... 71
2.12
Reviews
.................................. 72
2.12.1
Gaussian elimination and matrix decompositions
......... 73
2.12.2
Polynomial interpolation and divided differences
......... 74
VII
viii Contents
2.12.3 Orthogonal
and trigonometric
polynomials
............ 77
2.12.4
Basic
quadrature rules
....................... 79
2.12.5
Fixed point iteration and Newton s method
............ 80
2.12.6
Discrete and fast Fourier transforms
................ 82
2.13
Exercises
.................................. 83
3
Finite Difference and Finite Volume Methods
91
3.1
Semi-Discretization
............................ 92
3.1.1
Accuracy and derivation of spatial discretizations
......... 94
3.1.2
Staggered meshes
.......................... 98
3.1.3
Boundary conditions
........................ 106
3.1.4
The finite element method
..................... 110
3.1.5
Nonuniform
meshes
........................ 113
3.1.6
Stability and convergence
..................... 120
3.2
Full Discretization
............................. 120
3.2.1
Order, stability, and convergence
.................. 122
3.2.2
General linear stability
....................... 128
3.3
Exercises
.................................. 130
4
Stability for Constant Coefficient Problems
135
4.1
Fourier Analysis
.............................. 135
4.1.1
Stability for scalar equations
.................... 137
4.1.2
Stability for systems of equations
................. 139
4.1.3
Semi-discretization stability
.................... 142
4.1.4
Fourier analysis and ODE absolute stability regions
........ 143
4.2
Eigenvalue Analysis
............................ 144
4.3
Exercises
.................................. 146
5
Variable Coefficient and Nonlinear Problems
151
5.1
Freezing Coefficients and
Dissipati
vity..................
153
5.2
Schemes for Hyperbolic Systems in One Dimension
...........154
5.2.1
Lax-Wendroff and variants for conservation laws
.........156
5.2.2
Leapfrog and Lax-Friedriehs
...................158
5.2.3
Upwind scheme and the modified PDE
..............162
5.2.4
Box and Crank-Nicolson
......................165
5.3
Nonlinear Stability and Energy Methods
.................168
5.3.1
Energy method
...........................169
5.3.2
Runge-Kutta for skew-symmetric semi-discretizations
......173
5.4
Exercises
..................................177
6
Hamiltonian Systems and Long Time Integration
181
6.1
Hamiltonian Systems
............................182
6.2
Symplectic and Other Relevant Methods
.................185
6.2.1
Symplectic Runge-Kutta methods
.................188
6.2.2
Splitting and composition methods
................189
6.2.3
Variational methods
........................194
Contents
¡χ
6.3
Properties of Symplectic Methods
.....................195
6.4
Pitfalls in Highly Oscillatory Hamiltonian Systems
............198
6.5
Exercises
..................................205
7
Dispersion and Dissipation
211
7.1
Dispersion
.................................212
7.2
The Wave Equation
............................217
7.3
The KdV Equation
.............................230
7.3.1
Schemes based on a classical semi-discretization
.........232
7.3.2
Box schemes
............................236
7.4
Spectral Methods
..............................242
7.5
Lagrangian methods
............................246
7.6
Exercises
..................................246
8
More on Handling Boundary Conditions
253
8.1
Parabolic Problems
.............................253
8.2
Hyperbolic Problems
............................257
8.2.1
Boundary conditions for hyperbolic problems
...........257
8.2.2
Boundary conditions for discretized hyperbolic problems
.....261
8.2.3
Order reduction for Runge-Kutta methods
............268
8.3
Infinite or Large Domains
.........................271
8.4
Exercises
..................................272
9
Several Space Variables and Splitting Methods
275
9.1
Extending the Methods We Already Know
................276
9.2
Solving for Implicit Methods
.......................279
9.2.1
Implicit methods for parabolic equations
.............282
9.2.2
Alternating direction implicit methods
...............290
9.2.3
Nonlinear problems
........................292
9.3
Splitting Methods
.............................292
9.3.1
More general splitting
.......................297
9.3.2
Additive methods
..........................306
9.3.3
Exponential time differencing
...................310
9.4
Review: Iterative Methods for Linear Systems
..............312
9.4.1
Simplest iterative methods
.....................312
9.4.2
Conjugate gradient and related methods
..............314
9.4.3
Multigrid methods
.........................317
9.5
Exercises
..................................320
10
Discontinuities and Almost Discontinuities
327
10.1
Scalar Conservation Laws in One Dimension
...............329
10.1.1
Exact solution of the Riemann problem
..............333
10.2
First Order Schemes for Scalar Conservation Laws
............334
10.2.1
Godunov s scheme
.........................338
10.3
Higher Order Schemes for Scalar Conservation Laws
...........340
10.3.1
High-resolution schemes
......................341
x
Contents
10.3.2
Semi-discretization and
ENO
schemes
...............342
10.3.3
Strong stability preserving methods
................346
10.3.4
WENO
schemes
..........................349
10.4
Systems of Conservation Laws
......................352
10.5
Multidimensional Problems
........................356
10.6
Problems with Sharp Layers
........................358
10.7
Exercises
..................................360
11
Additional Topics
365
11.1
What First: Optimize or Discretize?
....................365
11.1.1
Symmetric matrices for
nonuniform
spatial meshes
........366
11.1.2
Efficient multigrid and Neumann BCs
...............366
11.1.3
Optimal control
...........................367
11.2
Nonuniform
Meshes
............................368
11.2.1
Adaptive meshes for steady state problems
............369
11.2.2
Adaptive mesh refinement
.....................371
11.2.3
Moving meshes
...........................371
11.3
Level Set Methods
.............................372
Bibliography
375
Index
387
|
| adam_txt |
Contents
Preface
χι
Introduction
1
1.1
Well-Posed
Initial Value
Problems. 4
1.1.1
Simple
model
cases
. 7
1.1.2
More general
cases
. 10
1.1.3
Initial
-boundary
value
problems
. 11
1.1.4
The solution
operator
. 12
1.2
A
Taste of Finite Differences.
12
1.2.1
Stability ideas
. 17
1.3
Reviews
. 24
1.3.1
Taylor's theorem
. 24
1.3.2
Matrix norms and eigenvalues
. 25
1.3.3
Function spaces
. 28
.3.4
The continuous Fourier transform
. 29
.3.5
The matrix power and exponential
. 30
.3.6
Fourier transform for periodic functions
. 31
1.4
Exercises
. 32
Methods and Concepts for ODEs
37
2.1
Linear Multistep Methods
. 39
2.2
Runge-Kutta Methods
. 42
2.3
Convergence and O-stability
. 48
2.4
Error Control and Estimation
. 52
2.5
Stability of ODE Methods
. 53
2.6
Stiffness
. 55
2.7
Solving Equations for Implicit Methods
. 59
2.8
Differential-Algebraic Equations
. 64
2.9
Symmetric and One-Sided Methods
. 66
2.10
Highly Oscillatory Problems
. 66
2.11
Boundary Value ODEs
. 71
2.12
Reviews
. 72
2.12.1
Gaussian elimination and matrix decompositions
. 73
2.12.2
Polynomial interpolation and divided differences
. 74
VII
viii Contents
2.12.3 Orthogonal
and trigonometric
polynomials
. 77
2.12.4
Basic
quadrature rules
. 79
2.12.5
Fixed point iteration and Newton's method
. 80
2.12.6
Discrete and fast Fourier transforms
. 82
2.13
Exercises
. 83
3
Finite Difference and Finite Volume Methods
91
3.1
Semi-Discretization
. 92
3.1.1
Accuracy and derivation of spatial discretizations
. 94
3.1.2
Staggered meshes
. 98
3.1.3
Boundary conditions
. 106
3.1.4
The finite element method
. 110
3.1.5
Nonuniform
meshes
. 113
3.1.6
Stability and convergence
. 120
3.2
Full Discretization
. 120
3.2.1
Order, stability, and convergence
. 122
3.2.2
General linear stability
. 128
3.3
Exercises
. 130
4
Stability for Constant Coefficient Problems
135
4.1
Fourier Analysis
. 135
4.1.1
Stability for scalar equations
. 137
4.1.2
Stability for systems of equations
. 139
4.1.3
Semi-discretization stability
. 142
4.1.4
Fourier analysis and ODE absolute stability regions
. 143
4.2
Eigenvalue Analysis
. 144
4.3
Exercises
. 146
5
Variable Coefficient and Nonlinear Problems
151
5.1
Freezing Coefficients and
Dissipati
vity.
153
5.2
Schemes for Hyperbolic Systems in One Dimension
.154
5.2.1
Lax-Wendroff and variants for conservation laws
.156
5.2.2
Leapfrog and Lax-Friedriehs
.158
5.2.3
Upwind scheme and the modified PDE
.162
5.2.4
Box and Crank-Nicolson
.165
5.3
Nonlinear Stability and Energy Methods
.168
5.3.1
Energy method
.169
5.3.2
Runge-Kutta for skew-symmetric semi-discretizations
.173
5.4
Exercises
.177
6
Hamiltonian Systems and Long Time Integration
181
6.1
Hamiltonian Systems
.182
6.2
Symplectic and Other Relevant Methods
.185
6.2.1
Symplectic Runge-Kutta methods
.188
6.2.2
Splitting and composition methods
.189
6.2.3
Variational methods
.194
Contents
¡χ
6.3
Properties of Symplectic Methods
.195
6.4
Pitfalls in Highly Oscillatory Hamiltonian Systems
.198
6.5
Exercises
.205
7
Dispersion and Dissipation
211
7.1
Dispersion
.212
7.2
The Wave Equation
.217
7.3
The KdV Equation
.230
7.3.1
Schemes based on a classical semi-discretization
.232
7.3.2
Box schemes
.236
7.4
Spectral Methods
.242
7.5
Lagrangian methods
.246
7.6
Exercises
.246
8
More on Handling Boundary Conditions
253
8.1
Parabolic Problems
.253
8.2
Hyperbolic Problems
.257
8.2.1
Boundary conditions for hyperbolic problems
.257
8.2.2
Boundary conditions for discretized hyperbolic problems
.261
8.2.3
Order reduction for Runge-Kutta methods
.268
8.3
Infinite or Large Domains
.271
8.4
Exercises
.272
9
Several Space Variables and Splitting Methods
275
9.1
Extending the Methods We Already Know
.276
9.2
Solving for Implicit Methods
.279
9.2.1
Implicit methods for parabolic equations
.282
9.2.2
Alternating direction implicit methods
.290
9.2.3
Nonlinear problems
.292
9.3
Splitting Methods
.292
9.3.1
More general splitting
.297
9.3.2
Additive methods
.306
9.3.3
Exponential time differencing
.310
9.4
Review: Iterative Methods for Linear Systems
.312
9.4.1
Simplest iterative methods
.312
9.4.2
Conjugate gradient and related methods
.314
9.4.3
Multigrid methods
.317
9.5
Exercises
.320
10
Discontinuities and Almost Discontinuities
327
10.1
Scalar Conservation Laws in One Dimension
.329
10.1.1
Exact solution of the Riemann problem
.333
10.2
First Order Schemes for Scalar Conservation Laws
.334
10.2.1
Godunov's scheme
.338
10.3
Higher Order Schemes for Scalar Conservation Laws
.340
10.3.1
High-resolution schemes
.341
x
Contents
10.3.2
Semi-discretization and
ENO
schemes
.342
10.3.3
Strong stability preserving methods
.346
10.3.4
WENO
schemes
.349
10.4
Systems of Conservation Laws
.352
10.5
Multidimensional Problems
.356
10.6
Problems with Sharp Layers
.358
10.7
Exercises
.360
11
Additional Topics
365
11.1
What First: Optimize or Discretize?
.365
11.1.1
Symmetric matrices for
nonuniform
spatial meshes
.366
11.1.2
Efficient multigrid and Neumann BCs
.366
11.1.3
Optimal control
.367
11.2
Nonuniform
Meshes
.368
11.2.1
Adaptive meshes for steady state problems
.369
11.2.2
Adaptive mesh refinement
.371
11.2.3
Moving meshes
.371
11.3
Level Set Methods
.372
Bibliography
375
Index
387 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ascher, Uri M. 1946- |
| author_GND | (DE-588)136140823 |
| author_facet | Ascher, Uri M. 1946- |
| author_role | aut |
| author_sort | Ascher, Uri M. 1946- |
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| bvnumber | BV023388519 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)192109753 (DE-599)HBZHT015536614 |
| dewey-full | 003/.5 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003/.5 |
| dewey-search | 003/.5 |
| dewey-sort | 13 15 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| format | Book |
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| id | DE-604.BV023388519 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:18:57Z |
| indexdate | 2024-07-09T21:17:29Z |
| institution | BVB |
| isbn | 9780898716528 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016571494 |
| oclc_num | 192109753 |
| open_access_boolean | |
| owner | DE-703 DE-355 DE-BY-UBR DE-83 DE-29T DE-634 DE-824 DE-11 |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-83 DE-29T DE-634 DE-824 DE-11 |
| physical | XIII, 395 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Soc. for Industrial and Applied Math. |
| record_format | marc |
| series | Computational science & engineering |
| series2 | Computational science & engineering |
| spelling | Ascher, Uri M. 1946- Verfasser (DE-588)136140823 aut Numerical methods for evolutionary differential equations Uri M. Ascher Philadelphia, Pa. Soc. for Industrial and Applied Math. 2008 XIII, 395 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Computational science & engineering 5 Literaturverz. S. 375 - 386 Análise numérica larpcal Equações diferenciais parciais larpcal Équations d'évolution - Solutions numériques Evolution equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Zeitabhängigkeit (DE-588)4320088-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Zeitabhängigkeit (DE-588)4320088-6 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Computational science & engineering 5 (DE-604)BV022382702 5 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016571494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ascher, Uri M. 1946- Numerical methods for evolutionary differential equations Computational science & engineering Análise numérica larpcal Equações diferenciais parciais larpcal Équations d'évolution - Solutions numériques Evolution equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Zeitabhängigkeit (DE-588)4320088-6 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4012249-9 (DE-588)4320088-6 |
| title | Numerical methods for evolutionary differential equations |
| title_auth | Numerical methods for evolutionary differential equations |
| title_exact_search | Numerical methods for evolutionary differential equations |
| title_exact_search_txtP | Numerical methods for evolutionary differential equations |
| title_full | Numerical methods for evolutionary differential equations Uri M. Ascher |
| title_fullStr | Numerical methods for evolutionary differential equations Uri M. Ascher |
| title_full_unstemmed | Numerical methods for evolutionary differential equations Uri M. Ascher |
| title_short | Numerical methods for evolutionary differential equations |
| title_sort | numerical methods for evolutionary differential equations |
| topic | Análise numérica larpcal Equações diferenciais parciais larpcal Équations d'évolution - Solutions numériques Evolution equations Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Zeitabhängigkeit (DE-588)4320088-6 gnd |
| topic_facet | Análise numérica Equações diferenciais parciais Équations d'évolution - Solutions numériques Evolution equations Numerical solutions Numerisches Verfahren Differentialgleichung Zeitabhängigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016571494&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022382702 |
| work_keys_str_mv | AT ascherurim numericalmethodsforevolutionarydifferentialequations |