Difference schemes: an introduction to the underlying theory
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| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Book |
| Language: | English Russian |
| Published: |
Amsterdam u.a.
North-Holland
1987
|
| Series: | Studies in mathematics and its applications
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Originaltitel: Raznostnye schemi |
| Physical Description: | XVII, 489 Seiten 23 cm |
| ISBN: | 0444702334 |
Staff View
MARC
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| 100 | 1 | |a Godunov, Sergej K. |d 1929-2023 |e Verfasser |0 (DE-588)173205828 |4 aut | |
| 240 | 1 | 0 | |a Raznostnye schemy |
| 245 | 1 | 0 | |a Difference schemes |b an introduction to the underlying theory |c S. K. Godunov and V. S. Ryabenkii. English translation [from the Russian] by E.M. Gelbard |
| 264 | 1 | |a Amsterdam u.a. |b North-Holland |c 1987 | |
| 300 | |a XVII, 489 Seiten |c 23 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Studies in mathematics and its applications |v 19 | |
| 500 | |a Originaltitel: Raznostnye schemi | ||
| 650 | 4 | |a Équations aux différences | |
| 650 | 4 | |a Équations différentielles linéaires | |
| 650 | 4 | |a Difference equations | |
| 650 | 4 | |a Differential equations, Linear | |
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Record in the Search Index
| _version_ | 1804115126873227264 |
|---|---|
| adam_text | ix
TABLE OF CONTENTS
Preface v
Preface to the Second Edition vi
Introduction 1
PART 1
ORDINARY DIFFERENCE EQUATIONS
Chapter 1. Difference Equations of First and Second Order.
Examples of Difference Schemes 5
§ 1. Simplest difference equations 5
1. Difference equations (1). 2. Order of difference
equations (8). 3. General solution of difference
equations (8).
Problems 11
§ 2. Difference equation of first order 12
1. Fundamental solution (12). 2. Conditions governing
the boundedness of the fundamental solution (13).
3. Particular solution (14).
Problems 16
§ 3. Difference equation of second order 17
1. General solution of the homogeneous equation (17).
2. General solution of the inhomogeneous equation.
Fundamental solution (21). 3. Estimate of the fundamental
solution In terms of the coefficients of the difference
equation (26).
Problems 28
x Table of Contents
Chapter 2. Boundary—Value Problems for Equations of Second Order .... 31
§ 4. Formulation of problem. Good conditioning criteria 31
1. Formulation of problem (31). 2. Definition of a well
conditioned problem (32). 3. Sufficient condition for
a well conditioned problem (34). 4. Criterion for a well
conditioned boundary value problem with constant coef¬
ficients (35). 5. Criterion for a well conditioned
problem with variable coefficients (35). 6. Justifica¬
tion of the criterion for a well conditioned boundary
value problem with constant coefficients (37).
7. General boundary value problem for a system of
difference equations (42).
Problems 46
§ 5. Algorithm for the solution of boundary value problems
forward elimination, back substitution (FEBS) 47
1. Description of forward elimination, back,
substitution (FEBS) (47). 2. Example of a computation¬
ally unstable algorithm (50).
Problems 51
Chapter 3. Basis of the FEBS Method 53
§ 6. Properties of well conditioned boundary value problems .... 53
1. Bound for the solution of a boundary value problem
with perturbed coefficients (53). 2. Proof of the
criterion for good conditioning (57), 3. Properties
of a well conditioned problem (62).
§ 7. Basis for the FEBS method in well conditioned boundary
value problems 63
1. Bounds on the FEBS coefficients (63). 2. Estimate of
the influence on computational results of rounding
errors committed in the course of calculation (65).
Table of Contents xi
PART 2
DIFFERENCE SCHEMES FOR ORDINARY DIFFERENTIAL EQUATIONS
Chapter 4. Elementary Examples of Difference Schemes 71
§ 8. The concept of order of accuracy and approximation 71
1. Order of accuracy of a difference scheme (71).
2. Speed of convergence of the solution of the difference
equation (75). 3. Order of approximation (77).
$ 9. Unstable difference schemes 78
1. Techniques for approximating the derivative (78).
2. Example of an unstable difference scheme (79).
Chapter 5. ¦ Convergence of the Solutions of Difference Equations as a
Consequence of Approximation and Stability 83
5 10. Convergence of a difference scheme 83
1. Concept of a net and a net function (83). 2. Conver¬
gent difference schemes (88}. 3. Proof of convergence
of a difference scheme (91).
Problems 93
§ 11. Approximation of a differential boundary value problem
by a difference scheme 94
1. The residual 5f(h) (94). 2. Computation of the resi¬
dual (96). 3. Approximation of order hk (98).
4. Examples (99). 5. Splitting of difference schemes
into subsystems (102). 6. Replacement of derivatives
by difference expressions (105). 7. Other methods for
constructing difference schemes (108).
Problems 109
§ 12. Definition of the stability of a difference scheme.
Convergence as a consequence of approximation and
stability 109
1. Definition of stability (109). 2. Connection between
approximation, stability and convergence (112). 3. Con¬
vergent difference scheme for an integral equation (118).
§ 13. On the choice of a norm 120
xii Table of Contents
§ 14. Sufficient condition for stability of difference schemes
for the solution of the Cauchy problem 128
1. Introductory example (129). 2. Canonical form of a
difference scheme (130). 3. Stability viewed as the
boundedness of the norms of powers of the transition
operator (132). 4. Examples of investigations of sta¬
bility (134). 5. Non uniqueness of the canonical
form (141).
Problems 143
§ 15. Necessary spectral criterion for stability 144
1. Boundedness of the norms of powers of the transition
operator necessary for stability (145). 2. Spectral cri¬
terion for stability (146). 3. Discussion of the spectral
stability criterion (147).
Problems 153
§ 16. Roundoff errors 154
1. Errors in the coefficients (154). 2. Computational
errors (157).
§ 17. Quantitative aspects of stability 159
§ 18. Method for studying stability of nonlinear problems 166
Chapter 6. Widely Used Difference Schemes 169
§ 19. Runga Kutta and Adams schemes 169
1. Runga Kutta scheme (170). 2. Adams schemes (172).
3. Note on stability (176). 4. Generalization to systems
of equations (177).
§ 20. Methods of solution for boundary value problems 179
1. The shooting method (180). 2. The FEBS method (182).
3. Newton s method (183).
Table of Contents xiii
PART 3
DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS. BASIC CONCEPTS
Chapter 7. Simplest Examples of the Construction and Study of
Difference Schemes 185
§ 21. Review and illustrations of basic definitions 185
1. Definition of convergence (185). 2. Definition of
approximation (186). 3. Definition of stability (190).
Problems 197
§ 22. Simplest methods for the construction of approximating
difference schemes 198
1. Replacement of derivatives by difference relations
(198). 2. Method of undetermined coefficients (206).
3. Schemes with recomputation, or predictor corrector
schemes (217). 4. On other examples (219).
Problems 219
§ 23. Examples of the formulation of boundary conditions in
the construction of difference schemes 221
Problems 226
§ 24. The Courant Friedrichs Levy condition, necessary for
convergence 228
1. The Courant Friedrichs Levy condition (228).
2. Examples of difference schemes for the Cauchy
problem (229). 3. Examples of difference schemes for
the Dirichlet problem (235).
Problems 238
Chapter 8. Some basic methods for the study of stability 241
§ 25. Spectral analysis of the Cauchy difference problem 241
1. Stability with respect to starting values (241).
2. Necessary spectral condition for stability (242).
3. Examples (244). 4. Integral representation of the
solution (252). 5. Smoothing of the difference solution
as a result of approximational viscosity (257).
Problems 260
xiv Table of Contents
§ 26. Principle of frozen coefficients 261
1. Frozen coefficients at interior points (262).
2. Criterion of Babenko and Gelfand (264).
Problems 270
§ 27. Representation of the solution of some model problems in
the form of finite Fourier series 272
1. Fourier series for net functions (272). 2. Represen¬
tation of the solutions of difference schemes for the
heat equation on an interval (276). 3. Representation of
the solution of difference schemes for the two
dimensional heat conduction problem (279). 4. Represen¬
tation of the solution of a difference scheme for the
vibrating string problem (282).
Problems 285
§ 28. The maximum principle 286
1. Explicit difference scheme (286). 2. Implicit
difference scheme (289). 3. Comparison of explicit and
implicit difference schemes (291).
Chapter 9. Difference Schene Concepts In the Computation
of Generalized Solutions 293
§ 29. The generalized solution 293
1. Mechanism generating discontinuities (294).
2. Definition of the generalized solution (295).
3. Condition on a line of discontinuity of a solution
(296). 4. Decay of an arbitrary discontinuity (298).
5. Other definitions of the generalized solution (299).
§ 30. The construction of difference schemes 300
1. Schemes with artificial viscosity (301). 2. Method
of characteristics (301). 3. Divergence difference
schemes (303).
Table of Contents xv
PART 4
PROBLEMS WITH TWO SPACE VARIABLE
Chapter 10. The Concept of Difference Schemes with Splitting 309
§ 31. Construction of splitting schemes 309
Problems 314
§ 32. Economical difference schemes 314
Problems 323
§ 33. Splitting by physical factors 323
Chapter 11. Elliptic problems 325
§ 34. Simplest difference scheme for the Dirichlet problem 325
1. Approximation (326). 2. Stability (327).
Problems 331
§ 35. Method of time development 332
1. Idea of the method of time development (332).
2. Analysis of the explicit time development scheme (335).
3. The alternating direction scheme (338). 4. Choice
of accuracy (340). 5. Limits of applicability of
methods (340).
Problems 341
§ 36. Iteration with variable step size 341
1. The idea of Richardson (341). 2. The Chebyshev
set of parameters (342). 3. Numbering of iteration
parameters (346). 4. The Douglas Rachford method (349).
Problems 352
§ 37. The Federenko method 353
1. Idea of the method (354). 2. Description of the
algorithm (356).
xvi Table of Contents
Chapter 12. Concept of Variational—Difference and Projection
Difference Schemes 357
§ 38. Variational and projection methods 357
1. Variational formulation of boundary value problems
(357). 2. Convergence of minimizing sequences (361).
3. The variational method of Ritz (365). 4. Projection
method of Galerkin (371). 5. Methods for solving the
algebraic system (373). 6. Computational stability (373).
Problems 374
§ 39. Construction and properties of variational difference
and projection difference schemes 375
1. Definition of variational difference and projection
difference schemes (375). 2. Example of a variatlonal
difference scheme for the first boundary value problem
(376). 3. An example of a variational difference scheme
for the third boundary value problem (385). 4. On the
method for proving convergence (388). 5. Comparison of
variational difference schemes with general variational
and ordinary difference schemes (389).
Problems 390
PART S
STABILITY OF EVOLUTIONAL BOUNDARY VALUE PROBLEMS VIEWED
AS THE BOUNDEDNESS OF NORMS OF POWERS OF A CERTAIN OPERATOR
Chapter 13. Construction of the Transition Operator 392
§ 40. Level structure of the solution of evolutional
problems 392
Problems 395
§ 41. Statement of the difference boundary—value problem in
the form up+1 = RhuP + xpp 396
1. Canonical form (396). 2. Stability as the uniform
boundedness of the norms of powers of R, (400).
3. Example (403).
Problems 406
§ 42. Use of particular solutions in the construction of
the transition operator 408
Table of Contents xvii
§ 43. Some methods for bounding norms of powers of operators .... 421
1. Necessary spectral conditions for the boundedness of
||RP|| (422). 2. Spectral criterion for the bounded
ness of powers of a selfadjoint operator (424). 3. Self
adiointness criteria (425). 4. Bounds on the eigen¬
values of operator R (426). 5. Choice of a scalar
product (428). 6. The stability criterion of Samarskii
(429).
Problems 431
Chapter 14. Spectral Criterion for the Stability of Nonselfadjoint
Evolutional Boundary Value Problems 433
§ 44. Spectrum of a family of operators JR } 433
1. Need for improvement in the spectral stability
criterion (433). 2. Definition of the spectrum of a
family of operators (435). 3. Necessary condition for
stability (436). 4. Discussion of the concept of the
spectrum of a family of operators {R,} (437). 5. Near¬
ness of the necessary stability criterion to sufficiency
(439).
§ 45. Algorithm for the computation of the spectrum of a
family of difference operators on net functions in an
interval 441
1. Typical example (441). 2. Algorithm for computing
the spectrum in the general case (449).
Problems 450
§ 46. The kernels of the spectra of families of operators 451
§ 47. On the staVxlity of iterative algorithms for the solu¬
tion of nonselfadjoint difference equations 455
Appendix. Method of internal boundary conditions 461
1. Class of systems of difference equations (461).
2. Fundamental solution (462). 3. Boundary of net
region (463). 4. Difference analogs of Cauchy and
Cauchy type integral formulas (464). 5. Internal
boundary conditions (466). 6. Boundary projection
operator (466). 7. General boundary value problem (467).
8. Basic idea of the method of internal boundary condi¬
tions (467). 9. Stability of internal boundary con¬
ditions (468). 10. Supplementary idea (469).
11. Comparison of the method of internal boundary
conditions with the method of singular integral
equations (470).
Bibliographical commentaries 475
Bibliography 483
Index 485
|
| any_adam_object | 1 |
| author | Godunov, Sergej K. 1929-2023 Rjabenʹkij, Viktor Solomonovič 1923- |
| author2 | Gelbard, Ely M. 1924- |
| author2_role | trl |
| author2_variant | e m g em emg |
| author_GND | (DE-588)173205828 (DE-588)1028057067 (DE-588)1301033855 |
| author_facet | Godunov, Sergej K. 1929-2023 Rjabenʹkij, Viktor Solomonovič 1923- Gelbard, Ely M. 1924- |
| author_role | aut aut |
| author_sort | Godunov, Sergej K. 1929-2023 |
| author_variant | s k g sk skg v s r vs vsr |
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| ctrlnum | (OCoLC)15519911 (DE-599)BVBBV000678692 |
| dewey-full | 515/.625 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.625 |
| dewey-search | 515/.625 |
| dewey-sort | 3515 3625 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV000678692 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:17:40Z |
| institution | BVB |
| isbn | 0444702334 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000423343 |
| oclc_num | 15519911 |
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| physical | XVII, 489 Seiten 23 cm |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | North-Holland |
| record_format | marc |
| series | Studies in mathematics and its applications |
| series2 | Studies in mathematics and its applications |
| spelling | Godunov, Sergej K. 1929-2023 Verfasser (DE-588)173205828 aut Raznostnye schemy Difference schemes an introduction to the underlying theory S. K. Godunov and V. S. Ryabenkii. English translation [from the Russian] by E.M. Gelbard Amsterdam u.a. North-Holland 1987 XVII, 489 Seiten 23 cm txt rdacontent n rdamedia nc rdacarrier Studies in mathematics and its applications 19 Originaltitel: Raznostnye schemi Équations aux différences Équations différentielles linéaires Difference equations Differential equations, Linear Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differenzengleichung (DE-588)4012264-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Differenzengleichung (DE-588)4012264-5 s DE-604 Differenzenverfahren (DE-588)4134362-1 s Rjabenʹkij, Viktor Solomonovič 1923- Verfasser (DE-588)1028057067 aut Gelbard, Ely M. 1924- (DE-588)1301033855 trl Studies in mathematics and its applications 19 (DE-604)BV000000646 19 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000423343&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Godunov, Sergej K. 1929-2023 Rjabenʹkij, Viktor Solomonovič 1923- Difference schemes an introduction to the underlying theory Studies in mathematics and its applications Équations aux différences Équations différentielles linéaires Difference equations Differential equations, Linear Differenzenverfahren (DE-588)4134362-1 gnd Differentialgleichung (DE-588)4012249-9 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| subject_GND | (DE-588)4134362-1 (DE-588)4012249-9 (DE-588)4012264-5 |
| title | Difference schemes an introduction to the underlying theory |
| title_alt | Raznostnye schemy |
| title_auth | Difference schemes an introduction to the underlying theory |
| title_exact_search | Difference schemes an introduction to the underlying theory |
| title_full | Difference schemes an introduction to the underlying theory S. K. Godunov and V. S. Ryabenkii. English translation [from the Russian] by E.M. Gelbard |
| title_fullStr | Difference schemes an introduction to the underlying theory S. K. Godunov and V. S. Ryabenkii. English translation [from the Russian] by E.M. Gelbard |
| title_full_unstemmed | Difference schemes an introduction to the underlying theory S. K. Godunov and V. S. Ryabenkii. English translation [from the Russian] by E.M. Gelbard |
| title_short | Difference schemes |
| title_sort | difference schemes an introduction to the underlying theory |
| title_sub | an introduction to the underlying theory |
| topic | Équations aux différences Équations différentielles linéaires Difference equations Differential equations, Linear Differenzenverfahren (DE-588)4134362-1 gnd Differentialgleichung (DE-588)4012249-9 gnd Differenzengleichung (DE-588)4012264-5 gnd |
| topic_facet | Équations aux différences Équations différentielles linéaires Difference equations Differential equations, Linear Differenzenverfahren Differentialgleichung Differenzengleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000423343&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
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