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Bibliographic Details
Main Authors: Zlatev, Zahari 1939- (Author), Dimov, Ivan 1952- (Author), Faragó, István 1950- (Author), Havasi, Ágnes (Author)
Format: Book
Language:English
Published: Berlin De Gruyter [2018]
Series:De Gruyter series in applied and numerical mathematics Volume 2
Subjects:
Richardson-Extrapolation
Runge-Kutta-Verfahren
Extrapolation
Differentialgleichung
Operator-Splitting-Verfahren
Numerisches Verfahren
Links: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
Physical Description:xvii, 292 Seiten Diagramme
ISBN:9783110516494
9783110533019
Staff View

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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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author Zlatev, Zahari 1939-
Dimov, Ivan 1952-
Faragó, István 1950-
Havasi, Ágnes
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Havasi, Ágnes
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series2 De Gruyter series in applied and numerical mathematics
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
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AT faragoistvan richardsonextrapolationpracticalaspectsandapplications
AT havasiagnes richardsonextrapolationpracticalaspectsandapplications
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