Difference methods for singular perturbation problems:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2009
|
| Series: | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics
140 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | "A Chapman & Hall book." Includes bibliographical references (p. 371-388) and index |
| Physical Description: | xv, 393 S. 25 cm |
| ISBN: | 9781584884590 1584884592 |
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MARC
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| 100 | 1 | |a Shishkin, Grigory I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Difference methods for singular perturbation problems |c Grigory I. Shishkin ; Lidia P. Shishkina |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2009 | |
| 300 | |a xv, 393 S. |c 25 cm | ||
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| 490 | 1 | |a Chapman & Hall/CRC monographs and surveys in pure and applied mathematics |v 140 | |
| 500 | |a "A Chapman & Hall book." | ||
| 500 | |a Includes bibliographical references (p. 371-388) and index | ||
| 650 | 4 | |a Singular perturbations (Mathematics) | |
| 650 | 4 | |a Difference equations |x Numerical solutions | |
| 650 | 4 | |a Algebra, Abstract | |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Shishkina, Lidia P. |e Verfasser |4 aut | |
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Record in the Search Index
| _version_ | 1804140695828561920 |
|---|---|
| adam_text | Contents
Preface
хш
I Grid approximations of singular perturbation
partial differential equations
1
1
Introduction
3
1.1
The development of numerical methods for singularly
perturbed problems
...................... 3
1.2
Theoretical problems in the construction of difference schemes
6
1.3
The main principles in the construction of special schemes
8
1.4
Modern trends in the development of special difference
schemes
............................. 10
1.5
The contents of the present book
............... 11
1.6
The present book
....................... 12
1.7
The audience for this book
.................. 16
2
Boundary value problems for elliptic reaction-diffusion
equations in domains with smooth boundaries
17
2.1
Problem formulation. The aim of the research
....... 17
2.2
Estimates of solutions and derivatives
............ 19
2.3
Conditions ensuring
ε
-uniform
convergence of difference
schemes for the problem on a slab
.............. 26
2.3.1
Sufficient conditions for
ε
-uniform
convergence of
difference schemes
................... 26
2.3.2
Sufficient conditions for
ε
-uniform
approximation of
the boundary value problem
.............. 29
2.3.3
Necessary conditions for distribution of mesh points
for
ε
-uniform
convergence of difference schemes.
Construction of condensing meshes
.......... 33
2.4
Monotone finite difference approximations of the boundary
value problem on a slab,
čľ-uniformly
convergent difference
schemes
............................. 38
2.4.1
Problems on uniform meshes
............. 38
2.4.2
Problems on piecewise-uniform meshes
........ 44
2.4.3
Consistent grids on
subdomains
............ 51
2.4.4
Є
-uniformly
convergent difference schemes
...... 57
2.5
Boundary value problems in domains with curvilinear
boundaries
........................... 58
2.5.1
A domain-decomposition-based difference scheme for
the boundary value problem on a slab
........ 58
2.5.2
A difference scheme for the boundary value problem
in a domain with curvilinear boundary
........ 67
3
Boundary value problems for elliptic reaction-diffusion
equations in domains with piecewise-smooth boundaries
75
3.1
Problem formulation. The aim of the research
....... 75
3.2
Estimates of solutions and derivatives
............ 76
3.3
Sufficient conditions for
ε
-uniform
convergence of a difference
scheme for the problem on a parallelepiped
......... 85
3.4
A difference scheme for the boundary value problem on
a parallelepiped
........................ 89
3.5
Consistent grids on
subdomains
............... 97
3.6
A difference scheme for the boundary value problem in
a domain with piecewise-uniform boundary
......... 102
4
Generalizations for elliptic reaction-diffusion equations
109
4.1
Monotonicity
of continual and discrete Schwartz methods
. 109
4.2
Approximation of the solution in a bounded
subdomain
for
the problem on a strip
..................... 112
4.3
Difference schemes of improved accuracy for the problem on
a slab
.............................. 120
4.4
Domain-decomposition method for improved iterative
schemes
............................. 125
5
Parabolic reaction-diffusion equations
133
5.1
Problem formulation
...................... 133
5.2
Estimates of solutions and derivatives
............ 134
5.3
Є-
uniformly convergent difference schemes
......... 145
5.3.1
Grid approximations of the boundary value problem
146
5.3.2
Consistent grids on a slab
............... 147
5.3.3
Consistent grids on a parallelepiped
......... 154
5.4
Consistent grids on
subdomains
............... 158
5.4.1
The problem on a slab
................. 158
5.4;2 The problem on a parallelepiped
........... 161
6
Elliptic convection-diffusion equations
165
6.1
Problem formulation
...................... 165
6.2
Estimates of solutions and derivatives
............ 166
6.2.1
The problem solution on a slab
............ 166
6.2.2
The problem on a parallelepiped
........... 169
6.3
On construction of
ε
-uniformly
convergent difference schemes
under their monotonicity condition
.............. 176
6.3.1
Analysis of necessary conditions for
ε
-uniform
convergence of difference schemes
........... 177
6.3.2
The problem on a slab
................. 180
6.3.3
The problem on a parallelepiped
........... 183
6.4
Monotone
ε
-uniformly
convergent difference schemes
.... 185
7
Parabolic convection-diffusion equations
191
7.1
Problem formulation
...................... 191
7.2
Estimates of the problem solution on a slab
......... 192
7.3
Estimates of the problem solution on a parallelepiped
... 199
7.4
Necessary conditions for
ε
-uniform
convergence of
difference schemes
....................... 206
7.5
Sufficient conditions for
ε
-uniform
convergence of monotone
difference schemes
....................... 210
7.6
Monotone
ε
-uniformly
convergent difference schemes
.... 213
II Advanced trends in
č-uniformly
convergent
difference methods
219
8
Grid approximations of parabolic reaction-diffusion
equations with three perturbation parameters
221
8.1
Introduction
.......................... 221
8.2
Problem formulation. The aim of the research
....... 222
8.3
A priori estimates
....................... 224
8.4
Grid approximations of the initial-boundary value problem
230
9
Application of widths for construction of difference
schemes for problems with moving boundary layers
235
9.1
Introduction
.......................... 235
9.2
A boundary value problem for a singularly perturbed
parabolic reaction-diffusion equation
............. 237
9.2.1
Problem
(9.2), (9.1) .................. 237
9.2.2
Some definitions
.................... 238
9.2.3
The aim of the research
................ 240
9.3
A priori estimates
....................... 241
9.4
Classical finite difference schemes
.............. 243
9.5
Construction of
ε
-uniform
and almost
e-uniform
approximations to solutions of problem
(9.2), (9.1)..... 246
9.6
Difference scheme on a grid adapted in the moving
boundary layer
......................... 251
9.7
Remarks and generalizations
................. 254
10
High-order accurate numerical methods for singularly
perturbed problems
259
10.1
Introduction
.......................... 259
10.2
Boundary value problems for singularly perturbed parabolic
convection-diffusion equations with sufficiently smooth data
261
10.2.1
Problem with sufficiently smooth data
........ 261
10.2.2
A finite difference scheme on an arbitrary grid
.... 262
10.2.3
Estimates of solutions on uniform grids
....... 263
10.2.4
Special
є
-uniform
convergent finite difference scheme
263
10.2.5
The aim of the research
................ 264
10.3
A priori estimates for problem with sufficiently smooth data
265
10.4
The defect correction method
................. 266
10.5
The Richardson extrapolation scheme
............ 270
10.6
Asymptotic constructs
..................... 273
10.7
A scheme with improved convergence for finite values of
є
. 275
10.8
Schemes based on asymptotic constructs
.......... 277
10.9
Boundary value problem for singularly perturbed parabolic
convection-diffusion equation with piecewise-smooth initial
data
............................... 280
10.9.1
Problem
(10.56)
with piecewise-smooth initial data
. 280
10.9.2
The aim of the research
................ 281
10.10
A priori estimates for the boundary value problem
(10.56)
with piecewise-smooth initial data
.............. 282
10.11
Classical finite difference approximations
.......... 285
10.12
Improved finite difference scheme
............... 287
11
A finite difference scheme on
α
priori adapted grids for
a singularly perturbed parabolic convection-diffusion
equation
289
11.1
Introduction
.......................... 289
11.2
Problem formulation. The aim of the research
....... 290
11.3
Grid approximations on locally refined grids that are uniform
in subdomains
......................... 293
11.4
Difference scheme on
α
priori adapted grid
......... 297
11.5
Convergence of the difference scheme on
α
priori adapted grid
303
11.6
Appendix
............................ 307
12
On conditioning of difference schemes and their matrices
for singularly perturbed problems
309
12.1
Introduction
.......................... 309
12.2
Conditioning of matrices to difference schemes on piecewise-
uniform and uniform meshes. Model problem for ODE
... 311
12.3
Conditioning of difference schemes on uniform and piecewise-
uniform grids for the model problem
............. 316
12.4
On conditioning of difference schemes and their matrices for
a parabolic problem
...................... 323
13
Approximation of systems of singularly perturbed elliptic
reaction-diffusion equations with two parameters
327
13.1
Introduction
.......................... 327
13.2
Problem formulation. The aim of the research
....... 328
13.3
Compatibility conditions. Some a priori estimates
..... 330
13.4
Derivation of a priori estimates for the problem
(13.2)
under
the condition
(13.5) ...................... 333
13.5
A priori estimates for the problem
(13.2)
under the conditions
(13.4), (13.6) .......................... 341
13.6
The classical finite difference scheme
............. 343
13.7
The special finite difference scheme
............. 345
13.8
Generalizations
......................... 348
14
Survey
349
14.1
Application of special numerical methods to mathematical
modeling problems
....................... 349
14.2
Numerical methods for problems with piecewise-smooth and
nonsmooth boundary functions
................ 351
14.3
On the approximation of solutions and derivatives
..... 352
14.4
On difference schemes on adaptive meshes
.......... 354
14.5
On the design of constructive difference schemes for an elliptic
convection-diffusion equation in an unbounded domain
. . 357
14.5.1
Problem formulation in an unbounded domain. The
task of computing the solution in a bounded domain
357
14.5.2
Domain of essential dependence for solutions of the
boundary value problem
................ 359
14.5.3
Generalizations
..................... 363
14.6
Compatibility conditions for a boundary value problem on a
rectangle for an elliptic convection-diffusion equation with a
perturbation vector parameter
................ 364
14.6.1
Problem formulation
.................. 365
14.6.2
Compatibility conditions
................ 366
References
371
Index
389
Mathematics
DIFFERENCE METHODS FOR SINGULAR
PERTURBATION PROBLEMS
Difference Methods for Singular Perturbation Problems focuses on the
development of robust difference schemes for wide classes of boundary
value problems. It justifies the
ε
-uniform
convergence of these schemes and
surveys the latest approaches important for further progress in numerical
methods.
The first part of the book explores boundary value problems for elliptic and
parabolic reaction-diffusion and convection-diffusion equations in n-
dimensional domains with smooth and piecewise-smooth boundaries.
Containing material published mainly in the last four years, the second
section focuses on problems with boundary layers and additional singularities
generated by nonsmooth data, unboundedness of the domain, and the
perturbation vector parameter.
Co-authored by the creator of the Shishkin mesh, this book presents a
systematic, detailed development of approaches to construct
ε
uniformly
convergent finite difference schemes for broad classes of singularly perturbed
boundary value problems.
Features
•
Develops the foundations of the theory of robust numerical methods for
singularly perturbed boundary value problems for partial differential
equations
•
Presents techniques to construct and justify a priori estimates of solutions
to boundary value problems, difference approximations of boundary
value problems, and special piecewise-uniform grids
•
Considers boundary value problems for multidimensional elliptic and
parabolic reaction- and convection-diffusion equations
Explores modern trends in the development of robust difference methods
for singularly perturbed problems, including numerical methods of high-
order accuracy, conditioning of robust finite difference schemes, and more
|
| any_adam_object | 1 |
| author | Shishkin, Grigory I. Shishkina, Lidia P. |
| author_facet | Shishkin, Grigory I. Shishkina, Lidia P. |
| author_role | aut aut |
| author_sort | Shishkin, Grigory I. |
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| building | Verbundindex |
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| callnumber-raw | QC20.7.P47 |
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| callnumber-subject | QC - Physics |
| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)232327369 (DE-599)BVBBV035767907 |
| dewey-full | 515/.392 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.392 |
| dewey-search | 515/.392 |
| dewey-sort | 3515 3392 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035767907 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:04:04Z |
| institution | BVB |
| isbn | 9781584884590 1584884592 |
| language | English |
| lccn | 2008025636 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018627661 |
| oclc_num | 232327369 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | xv, 393 S. 25 cm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | CRC Press |
| record_format | marc |
| series | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics |
| series2 | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics |
| spelling | Shishkin, Grigory I. Verfasser aut Difference methods for singular perturbation problems Grigory I. Shishkin ; Lidia P. Shishkina Boca Raton [u.a.] CRC Press 2009 xv, 393 S. 25 cm txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 140 "A Chapman & Hall book." Includes bibliographical references (p. 371-388) and index Singular perturbations (Mathematics) Difference equations Numerical solutions Algebra, Abstract Randwertproblem (DE-588)4048395-2 gnd rswk-swf Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Differenzenverfahren (DE-588)4134362-1 s Randwertproblem (DE-588)4048395-2 s DE-604 Shishkina, Lidia P. Verfasser aut Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 140 (DE-604)BV013350872 140 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627661&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=018627661&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Shishkin, Grigory I. Shishkina, Lidia P. Difference methods for singular perturbation problems Chapman & Hall/CRC monographs and surveys in pure and applied mathematics Singular perturbations (Mathematics) Difference equations Numerical solutions Algebra, Abstract Randwertproblem (DE-588)4048395-2 gnd Differenzenverfahren (DE-588)4134362-1 gnd |
| subject_GND | (DE-588)4048395-2 (DE-588)4134362-1 |
| title | Difference methods for singular perturbation problems |
| title_auth | Difference methods for singular perturbation problems |
| title_exact_search | Difference methods for singular perturbation problems |
| title_full | Difference methods for singular perturbation problems Grigory I. Shishkin ; Lidia P. Shishkina |
| title_fullStr | Difference methods for singular perturbation problems Grigory I. Shishkin ; Lidia P. Shishkina |
| title_full_unstemmed | Difference methods for singular perturbation problems Grigory I. Shishkin ; Lidia P. Shishkina |
| title_short | Difference methods for singular perturbation problems |
| title_sort | difference methods for singular perturbation problems |
| topic | Singular perturbations (Mathematics) Difference equations Numerical solutions Algebra, Abstract Randwertproblem (DE-588)4048395-2 gnd Differenzenverfahren (DE-588)4134362-1 gnd |
| topic_facet | Singular perturbations (Mathematics) Difference equations Numerical solutions Algebra, Abstract Randwertproblem Differenzenverfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627661&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=018627661&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013350872 |
| work_keys_str_mv | AT shishkingrigoryi differencemethodsforsingularperturbationproblems AT shishkinalidiap differencemethodsforsingularperturbationproblems |