Richardson extrapolation: practical aspects and applications
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
De Gruyter
[2018]
|
| Series: | De Gruyter series in applied and numerical mathematics
Volume 2 |
| Subjects: | |
| Online Access: | http://www.degruyter.com/search?f_0=isbnissn&q_0=9783110516494&searchTitles=true Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | xvii, 292 Seiten Diagramme |
| ISBN: | 9783110516494 9783110533019 |
Staff View
MARC
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| 245 | 1 | 0 | |a Richardson extrapolation |b practical aspects and applications |c Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi |
| 264 | 1 | |a Berlin |b De Gruyter |c [2018] | |
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Record in the Search Index
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| adam_text | CONTENTS
PREFACE
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
2
2.1
2.2
2.3
2.4
2.4.1
2.4.2
2.4.3
2.4.4
2.4.5
2.5
2.5.1
2.5.2
2.5.3
2.6
2.7
2.8
THE BASIC PROPERTIES OF RICHARDSON EXTRAPOLATION - 1
THE INTRODUCTION OF AN INITIAL VALUE PROBLEM FOR SYSTEMS OF ODES - 3
NUMERICAL TREATMENT OF INITIAL VALUE PROBLEMS FOR SYSTEMS OF ODES -
AN INTRODUCTION TO RICHARDSON EXTRAPOLATION - 8
ACCURACY OF RICHARDSON EXTRAPOLATION - 9
EVALUATION OF THE ERROR -
10
MAJOR DRAWBACKS AND ADVANTAGES OF RICHARDSON EXTRAPOLATION -
11
TWO IMPLEMENTATIONS OF RICHARDSON EXTRAPOLATION -
17
OBTAINING HIGHER ACCURACY - 20
GENERAL CONCLUSIONS RELATED TO CHAPTER 1 - 21
TOPICS FOR FURTHER RESEARCH - 22
RICHARDSON EXTRAPOLATION FOR EXPLICIT RUNGE-KUTTA METHODS - 23
STABILITY FUNCTION OF ONE-STEP METHODS
FOR SOLVING SYSTEMS OF ODES - 25
STABILITY POLYNOMIALS OF EXPLICIT RUNGE-KUTTA METHODS - 30
USING RICHARDSON EXTRAPOLATION TOGETHER WITH
THE SCALAR TEST PROBLEM - 31
IMPACT OF RICHARDSON EXTRAPOLATION
ON ABSOLUTE STABILITY PROPERTIES - 32
STABILITY REGIONS RELATED TO THE FIRST-ORDER ONE-STAGE
EXPLICIT RUNGE-KUTTA METHOD - 34
STABILITY REGIONS RELATED TO THE SECOND-ORDER TWO-STAGE
EXPLICIT RUNGE-KUTTA METHODS
----
35
STABILITY REGIONS RELATED TO THIRD-ORDER THREE-STAGE
EXPLICIT RUNGE-KUTTA METHODS - 36
STABILITY REGIONS RELATED TO FOURTH-ORDER FOUR-STAGE
EXPLICIT RUNGE-KUTTA METHODS
----
38
ON THE USE OF COMPLEX ARITHMETIC IN THE PROGRAM
FOR DRAWING THE PLOTS
----
39
PREPARATION OF APPROPRIATE NUMERICAL EXAMPLES - 39
NUMERICAL EXAMPLE WITH A LARGE REAL EIGENVALUE - 40
NUMERICAL EXAMPLE WITH LARGE COMPLEX EIGENVALUES - 40
A NONLINEAR NUMERICAL EXAMPLE - 42
ORGANIZATION OF THE COMPUTATIONS - 44
PARTICULAR NUMERICAL METHODS USED IN THE EXPERIMENTS - 45
NUMERICAL RESULTS - 46
2.9
2.9.1
2.9.2
2.9.3
2.9.4
2.10
2.11
3
3.1
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
3.2
3.2.1
3.2.2
3.3
3.4
3.5
3.6
3.7
3.8
3.8.1
3.8.2
3.8.3
3.9
3.10
3.11
3.12
4
4.1
4.2
4.3
THE DEVELOPMENT OF METHODS WITH ENHANCED
ABSOLUTE STABILITY PROPERTIES - 55
THE DERIVATION OF TWO CLASSES OF NUMERICAL METHODS
WITH GOOD STABILITY PROPERTIES - 56
SELECTING PARTICULAR NUMERICAL METHODS FOR CASE 1:
P = 3
AND
M
= 4
----
60
SELECTING PARTICULAR NUMERICAL METHODS FOR CASE 2:
P = 4 AND
M
= 6
----
62
POSSIBILITIES FOR FURTHER IMPROVEMENT OF THE RESULTS - 68
GENERAL CONCLUSIONS RELATED TO CHAPTER 2
----
72
TOPICS FOR FURTHER RESEARCH - 74
LINEAR MULTISTEP AND PREDICTOR-CORRECTOR METHODS - 75
LINEAR MULTISTEP METHODS FOR SOLVING SYSTEMS OF ODES - 77
ORDER CONDITIONS
----
78
BASIC DEFINITIONS
----
78
ATTAINABLE ORDER OF LINEAR MULTISTEP METHODS - 80
DRAWBACKS AND ADVANTAGES OF LINEAR MULTISTEP METHODS - 80
FREQUENTLY USED LINEAR MULTISTEP METHODS - 82
VARIATION OF THE TIME-STEP SIZE FOR LINEAR MULTISTEP METHODS
----
83
CALCULATION OF THE COEFFICIENTS OF AN LM VSVFM
----
84
ZERO-STABILITY PROPERTIES OF AN LM VSVFM
----
85
ABSOLUTE STABILITY OF THE LINEAR MULTISTEP METHODS
----
86
DIFFICULTIES RELATED TO THE IMPLEMENTATION
OF RICHARDSON EXTRAPOLATION - 90
INTRODUCTION OF SOME PREDICTOR-CORRECTOR SCHEMES - 91
LOCAL ERROR ESTIMATION - 93
ABSOLUTE STABILITY OF PREDICTOR-CORRECTOR SCHEMES
----
98
THE APPLICATION OF SEVERAL DIFFERENT CORRECTORS -
102
AN EXAMPLE FROM THE FIELD OF ENVIRONMENTAL MODELLING
----
103
SOME ABSOLUTE STABILITY CONSIDERATIONS RELATED TO
ENVIRONMENTAL MODELLING -
105
NUMERICAL EXPERIMENTS
----
108
A-STABILITY OF THE LINEAR MULTISTEP METHODS
----
110
COEFFICIENTS OF SOME POPULAR LINEAR MULTISTEP METHODS
----
111
GENERAL CONCLUSIONS RELATED TO CHAPTER 3
----
111
TOPICS FOR FURTHER RESEARCH -
117
RICHARDSON EXTRAPOLATION FOR SOME IMPLICIT METHODS -
119
DESCRIPTION OF THE CLASS OF 6-METHODS
----
120
STABILITY PROPERTIES OF THE 6-METHODS -
122
COMBINING THE 6-METHOD WITH RICHARDSON EXTRAPOLATION
----
126
4.4
4.5
4.5.1
4.5.2
4.5.3
4.5.4
4.5.5
4.6
4.6.1
4.6.2
4.6.3
4.6.4
4.6.5
4.6.6
4.7
4.7.1
4.7.2
4.7.3
4.7.4
4.7.5
4.7.6
4.7.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.4.1
5.4.2
5.4.3
5.4.4
5.4.5
STABILITY OF RICHARDSON EXTRAPOLATION COMBINED WITH 6-METHOD -
127
THE PROBLEM WITH IMPLICITNESS -
134
APPLICATION OF THE CLASSICAL NEWTON ITERATIVE METHOD
----
135
APPLICATION OF THE MODIFIED NEWTON ITERATIVE METHOD -
138
ACHIEVING BETTER EFFICIENCY BY KEEPING AN OLD DECOMPOSITION
OF THE JACOBIAN MATRIX -
139
SELECTING STOPPING CRITERIA -
140
RICHARDSON EXTRAPOLATION AND THE NEWTON METHOD -
144
NUMERICAL EXPERIMENTS -
146
ATMOSPHERIC CHEMICAL SCHEME -
146
ORGANIZATION OF THE COMPUTATIONS -
149
ACHIEVING SECOND-ORDER ACCURACY -
152
COMPARISON OF THE 6-METHOD WITH 6 =
0.75
AND THE BACKWARD EULER FORMULA
----
153
COMPARING THE COMPUTING TIMES NEEDED TO OBTAIN
THE PRESCRIBED ACCURACY -
154
USING THE TRAPEZOIDAL RULE IN THE COMPUTATIONS -
157
USING IMPLICIT RUNGE-KUTTA METHODS
----
159
FULLY IMPLICIT RUNGE KUTTA METHODS
----
159
DIAGONALLY IMPLICIT RUNGE-KUTTA METHODS -
160
EVALUATING THE REDUCTION OF THE COMPUTATIONAL COST
WHEN DIRK METHODS ARE USED
----
161
APPLYING RICHARDSON EXTRAPOLATION FOR FULLY IMPLICIT
RUNGE-KUTTA METHODS
----
163
APPLYING RICHARDSON EXTRAPOLATION FOR DIAGONALLY IMPLICIT
RUNGE-KUTTA METHODS
----
166
STABILITY RESULTS RELATED TO ACTIVE RICHARDSON EXTRAPOLATION -
167
NUMERICAL EXPERIMENTS -
170
GENERAL CONCLUSIONS RELATED TO CHAPTER 4 -
178
TOPICS FOR FURTHER RESEARCH -
179
RICHARDSON EXTRAPOLATION FOR SPLITTING TECHNIQUES -
181
RICHARDSON EXTRAPOLATION FOR SEQUENTIAL SPLITTING -
181
DERIVATION OF THE STABILITY FUNCTION FOR THE SEQUENTIAL
SPLITTING PROCEDURE
----
183
STABILITY PROPERTIES OF THE SEQUENTIAL SPLITTING PROCEDURE -
186
SOME NUMERICAL EXPERIMENTS -
192
SPLITTING THE ATMOSPHERIC CHEMICAL SCHEME -
193
ORGANIZATION OF THE COMPUTATIONS
----
193
RESULTS OBTAINED WHEN THE BACKWARD EULER FORMULA IS USED -
195
RESULTS OBTAINED WHEN THE 6-METHOD WITH 6 =
0.75
IS USED
----
195
RESULTS OBTAINED WHEN THE TRAPEZOIDAL RULE IS USED -
197
5.4.6
5.5
5.5.1
5.5.2
5.5.3
5.5.4
5.5.5
5.5.6
5.5.7
5.6
5.7
6
6.1
6.2
6.3
6.4
6.5
6.5.1
6.5.2
6.5.3
6.5.4
6.5.5
6.6
6.6.1
6.6.2
6.6.3
6.6.4
6.6.5
6.7
6.8
7
7.1
SOME CONCLUSIONS FROM THE NUMERICAL EXPERIMENTS -
197
MARCHUK-STRANG SPLITTING PROCEDURE -
198
SOME INTRODUCTORY REMARKS
----
198
THE MARCHUK-STRANG SPLITTING PROCEDURE
AND RICHARDSON EXTRAPOLATION - 200
STABILITY FUNCTION OF THE COMBINED NUMERICAL METHOD - 201
SELECTION OF AN APPROPRIATE CLASS OF RUNGE-KUTTA METHODS
----
203
ABSOLUTE STABILITY REGIONS OF THE COMBINED NUMERICAL METHODS
----
205
SOME NUMERICAL RESULTS
----
207
CONCLUDING REMARKS - 213
GENERAL CONCLUSIONS RELATED TO CHAPTER 5 - 213
TOPICS FOR FURTHER RESEARCH - 214
RICHARDSON EXTRAPOLATION FOR ADVECTION PROBLEMS - 215
THE ONE-DIMENSIONAL ADVECTION PROBLEM - 216
COMBINING THE ADVECTION PROBLEM
WITH RICHARDSON EXTRAPOLATION
----
218
IMPLEMENTATION OF RICHARDSON EXTRAPOLATION - 220
ORDER OF ACCURACY OF THE COMBINED NUMERICAL METHOD
----
221
THREE NUMERICAL EXAMPLES - 225
ORGANIZATION OF THE COMPUTATIONS - 225
CONSTRUCTION OF A TEST PROBLEM WITH STEEP GRADIENTS
OF THE UNKNOWN FUNCTION
----
227
CONSTRUCTION OF AN OSCILLATORY TEST PROBLEM - 230
CONSTRUCTION OF A TEST PROBLEM WITH DISCONTINUOUS DERIVATIVES
OF THE UNKNOWN FUNCTION - 231
COMPARISON OF THE FOUR IMPLEMENTATIONS OF RICHARDSON
EXTRAPOLATION
----
233
MULTI DIMENSIONAL ADVECTION PROBLEM
----
235
INTRODUCTION OF THE MULTI DIMENSIONAL ADVECTION EQUATION - 235
EXPANDING THE UNKNOWN FUNCTION IN TAYLOR SERIES - 236
THREE SPECIAL CASES
----
241
DESIGNING A SECOND-ORDER NUMERICAL METHOD
FOR MULTI DIMENSIONAL ADVECTION - 243
APPLICATION OF RICHARDSON EXTRAPOLATION
----
244
GENERAL CONCLUSIONS RELATED TO CHAPTER 6
----
255
TOPICS FOR FURTHER RESEARCH - 256
RICHARDSON EXTRAPOLATION FOR SOME OTHER PROBLEMS - 257
ACCELERATION OF THE SPEED OF CONVERGENCE FOR SEQUENCES
OF REAL NUMBERS - 258
IMPROVING THE ACCURACY IN CALCULATIONS RELATED TO THE NUMBER N - 259
7.1.1
7.1.2 DEFINITIONS RELATED TO THE CONVERGENCE RATE
OF SOME ITERATIVE PROCESSES - 259
7.1.3 THE AITKEN SCHEME AND STEFFENSEN S IMPROVEMENT - 261
7.1.4 RICHARDSON EXTRAPOLATION FOR SEQUENCES OF REAL NUMBERS:
AN EXAMPLE - 264
7.2 APPLICATION OF RICHARDSON EXTRAPOLATION
TO NUMERICAL INTEGRATION - 268
7.3 GENERAL CONCLUSIONS RELATED TO CHAPTER 7 - 271
7.4 TOPICS FOR FURTHER RESEARCH - 271
8 GENERAL CONCLUSIONS - 277
REFERENCES - 279
LIST OF ABBREVIATIONS - 287
AUTHOR INDEX - 289
SUBJECT INDEX - 291
|
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| id | DE-604.BV044718598 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:00:15Z |
| institution | BVB |
| institution_GND | (DE-588)10095502-2 |
| isbn | 9783110516494 9783110533019 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030114947 |
| oclc_num | 1015870216 |
| open_access_boolean | |
| owner | DE-634 |
| owner_facet | DE-634 |
| physical | xvii, 292 Seiten Diagramme |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter series in applied and numerical mathematics |
| series2 | De Gruyter series in applied and numerical mathematics |
| spelling | Zlatev, Zahari 1939- Verfasser (DE-588)13264357X aut Richardson extrapolation practical aspects and applications Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi Berlin De Gruyter [2018] xvii, 292 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier De Gruyter series in applied and numerical mathematics Volume 2 Richardson-Extrapolation (DE-588)1146018282 gnd rswk-swf Runge-Kutta-Verfahren Extrapolation Differentialgleichung Operator-Splitting-Verfahren Numerisches Verfahren Richardson-Extrapolation (DE-588)1146018282 s DE-604 Dimov, Ivan 1952- Verfasser (DE-588)1043787461 aut Faragó, István 1950- Verfasser (DE-588)1053133243 aut Havasi, Ágnes Verfasser aut Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, PDF 978-3-11-053300-2 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-053301-9 De Gruyter series in applied and numerical mathematics Volume 2 (DE-604)BV044780807 2 X:MVB http://www.degruyter.com/search?f_0=isbnissn&q_0=9783110516494&searchTitles=true B:DE-101 application/pdf http://d-nb.info/1127760718/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030114947&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zlatev, Zahari 1939- Dimov, Ivan 1952- Faragó, István 1950- Havasi, Ágnes Richardson extrapolation practical aspects and applications De Gruyter series in applied and numerical mathematics Richardson-Extrapolation (DE-588)1146018282 gnd |
| subject_GND | (DE-588)1146018282 |
| title | Richardson extrapolation practical aspects and applications |
| title_auth | Richardson extrapolation practical aspects and applications |
| title_exact_search | Richardson extrapolation practical aspects and applications |
| title_full | Richardson extrapolation practical aspects and applications Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi |
| title_fullStr | Richardson extrapolation practical aspects and applications Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi |
| title_full_unstemmed | Richardson extrapolation practical aspects and applications Zahari Zlatev, Ivan Dimov, István Faragó, Ágnes Havasi |
| title_short | Richardson extrapolation |
| title_sort | richardson extrapolation practical aspects and applications |
| title_sub | practical aspects and applications |
| topic | Richardson-Extrapolation (DE-588)1146018282 gnd |
| topic_facet | Richardson-Extrapolation |
| url | http://www.degruyter.com/search?f_0=isbnissn&q_0=9783110516494&searchTitles=true http://d-nb.info/1127760718/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030114947&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044780807 |
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