Holdings: Applications of nonstandard finite difference schemes

Applications of nonstandard finite difference schemes:

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Bibliographic Details
Format:	Book
Language:	English
Published:	Singapore [u.a.] World Scientific 2000
Subjects:	Differential equations, Partial > Numerical solutions Finite differences Konferenzschrift
Online Access:	Inhaltsverzeichnis
Physical Description:	XII, 250 S. graph. Darst.
ISBN:	981024133X

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245	1	0	\|a Applications of nonstandard finite difference schemes \|c ed.: Ronald E. Mickens
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650		4	\|a Differential equations, Partial \|x Numerical solutions
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adam_text	CONT ENT S PREFACE CHAPTER 1 NONSTANDARD FINITE DIFFERENCE SCHEMES RONALD E . MAEKENS ............................. 1.1 INTRODUCTION ........................... 1.2 EXACT SCHEMES ....................... 1.3 NONSTANDARD SCHEMES ............................ 1.4 APPLICATIONS ................. 1.4.1 FIRST-ORDER SCALAR ODE S ................. 1.4.2 A PHOTOCONDUCTION MODEL ................... 1.4.3 THE DUFFING OSCILLATOR .................. 1.4.4 MIXED PARITY OSCILLATOR 1.4.5 A CUBIC REACTION PROBLEM IN NEUROPHYSIOLOGY ..... 1.4.6 TIME-INDEPENDENT SCHROEDINGER EQUATIONS ........ ................. 1.4.7 TRAVELING WAVE SOLUTIONS ........... 1.4.8 LINEAR ADVECTION-DIFFUSION EQUATION ................... 1.4.9 A COMBUSTION MODEL 1.4.10 INFLUENCE OF SPATIAL DISCRETIZATIONS FOR PDE S ...... .......................... 1.5 FUTURE DIRECTIONS BIBLIOGRAPHY CHAPTER 2 NONSTANDARD METHODS FOR ADVECTION-DIFFUSION- REACT ION EQUATIONS HRISTO V . KOJOUHAMV AND BENITO M . CHEN ............................. 2.1 INTRODUCTION X CONTENTS 2.2 NON-STANDARD METHODS IN ONE DIMENSION ............ 58 2.2.1 ADVECTION-FTEACTION EQUATIONS .............. 59 2.2.1.1 LOGISTIC GROWTH REACTION TERMS ........ 59 2.2.1.2 LINEAR REACTION TERMS ............. 62 2.2.1.3 NONLINEAR REACTION TERMS ........... 64 2.2.2 ADVECTION-DIFFUSION-REACTION EQUATIONS ......... 67 2.3 ERROR ANALYSIS OF THE NON-STANDARD METHOD ........... 69 2.3.1 ADVECTION-REACTION EQUATIONS .............. 69 2.3.1.1 ZERO LOCAL TIME- ILUNCATION ERROR ...... 70 2.3.1.2 ZERO LOCAL SPACETRUNCATION ERROR ...... 71 2.3.1.3 INTERPOLATION ERRORS ............... 72 2.3.2 ADVECTION-DIFFUSION-REACTION EQUATIONS ......... 77 2.4 ONE-DIMENSIONAL NUMERICAL FTESULTS ............... 84 2.4.1 ADVECTION-REACTION EQUATIONS .............. 85 2.4.2 ADVECTION-DIFFUSION-HRACTION EQUATIONS ......... 90 2.5 NON-STANDARD METHODS IN MULTIPLE DIMENSIONS ......... 94 2.5.1 DEVELOPMENT OF THE NON-STANDARD METHOD ....... 94 2.5.2 ERROR ESTIMATES ...................... 98 2.6 SUMMARY .............................. 104 BIBLIOGRAPHY 106 CHAPTER 3 APPLICATION OF NONSTANDARD FINITE DIFFERENCES TO SOLVE THE WAVE EQUATION AND MAXWELL S EQUATIONS 109 JAMES B . COLE 3.1 INTRODUCTION ............................. 109 3.2 THE ONE-DIMENSIONAL WAVE EQUATION .............. 111 3.2.1 FITE-DIFFERENTE TIME-DOMAIN ALGORITHM ........ 111 3.2.2 ALGORITHMIC ERROR ..................... 115 3.2.3 NONSTANDARD FITE DIFIERENCES .............. 117 3.2.4 ONEDIMENSIONAL SCATTERING ................ 120 3.3 THE TWO- AND THREE-DIMENSIONAL WAVE EQUATION ....... 124 3.3.1 STANDARD FDTD ALGORITHM ................ 124 3.3.2 GENERALIZED NONSTANDARD FINITE DIFFERENCES - TWO DIMENSIONS ......................... 125 3.2.3 GENERALIZED NONSTANDARD FINITE DIFFERENCES IN THREE DIMENSIONS ......................... 130 3.4 DISCRETIZATION AND STABILITY .................... 133 CONTENTS XI ........................ 3.4.1 DISCRETIZATION 133 ........................... 3.4.2 STABILITY 133 ............. 3.5 NSFD SOLUTION OF MAXWELL S EQUATIONS 139 3.5.1 THE STANDARD YEE ALGORITHM ............... 139 ................ 3.5.2 NONSTANDARD YEE ALGORITHRN 142 ..... 3.5.3 MAXWELL S EQUATIONS IN A CONDUCTING MEDIUM 146 ............... 3.5.4 ELECTROMAGNETIC SIMULATIONS 149 .............................. 3.6 SUMMARY 150 ............................. 3.7 CONCLUSION 150 BIBLIOGRAPHY 152 CHAPTER 4 NON-STANDARD DISCRETIZATION METHODS FOR SOME BIOLOGICAL MODELS 155 H . ALKAHBY, F . DANNAN. OND S . ELAYDI ............................. 4.1 INTRODUCTION 155 ........ 4.2 STABILITY OF LOTHVOLTERRA DIFFERENTIAL EQUATIONS 157 ...................... 4.3 CLASSICAL DISCRETIZATION 158 .............. 4.4 NONSTANDARD DISCRETIZATION SCHEMES 160 .......... 4.5 COMPETITIVE AND COOPERATIVE DISCRETE MODELS 161 4.6 PERMANENCE OF DISCRETE COMPETITIVE AND ........................ COOPERATIVE SYSTEMS 164 4.7 PREDATOR-PREY DISCRETE MODELS .................. 165 4.8 GLOBAL STABILITY OF COMPETITIVE AND ........................ COOPERATIVE SYSTEMS 170 4.9 LESLIE PREDATOR-PREY MODEL .................... 172 4.10 OTHER NONSTANDARD NUMERICAL SCHEMES ............. 174 4.11 A KOLMOGOROV MODEL OF COOPERATIVE SYSTEMS .......... 177 ........................... 4.12 OPEN PROBLEMS 178 BIBLIOGRAPHY 179 CHAPTER 5 AN INTRODUCTION TO NUMERICAL INTEGRATORS PRE- SERVING PHYSICAL PROPERTIES 181 MARTIN J . GANDER UND RITA MEYER-SPASCHE ............................. 5.1 INTRODUCTION 182 ..................... 5.2 EXACT DIFFERENTE SCHEMES 184 5.2.1 STANDARD NUMERICAL SCHEMES AS EXACT SCHEMES ..... 184 XII CONTENTS .............. 5.2.1.1 FIRST ORDER SCHEMES 185 ............. 5.2.1.2 SECOND ORDER SCHEMES 187 ............. 5.2.1.3 HIGHER ORDER SCHEMES 192 ............. 5.2.1.4 RUNGE-KUTTA SCHEMES 195 ............. 5.2.2 FUNCTIONAL FITTING RK-METHODS 199 5.2.3 SCHEMES FOR GIVEN DIFFERENTIAL EQUATIONS ........ 203 5.2.3.1 EXACT SCHEMES FOR GIVEN DIFFERENTIAL EQUATIONS 203 5.2.3.2 NONSTANDARD SCHEMES FOR PARABOLIC EQUATIONS WITH BLOW-UP SOLUTIONS: LE-ROUX SCHEMES . . 205 ................. 5.3 DYNAMICS OF DIFFERENCE SCHEMES 208 .............. 5.3.1 CONTINUOUS DYNAMICAL SYSTEMS 208 ............... 5.3.2 DISCRETE DYNAMICAL SYSTEMS 210 ................... 5.3.3 FORWARD EULER SCHEME 212 .................. 5.3.4 MIDPOINT EULER SCHEME 215 ............. 5.3.5 LIEARLY IMPLICIT EULER SCHEMES 219 ............ 5.3.5.1 DETAILS OF THE DYNAMICS 221 .................. 5.3.5.2 SUPERSTABILITY 223 .......... 5.3.6 THE LINEARLY HPLICIT LINTRAP SCHEME 224 ............... 5.3.6.1 EXISTENCE INTERVALS 226 ....... 5.3.6.2 ADJOINT AND SELF-ADJOINT SCHEMES 227 ........... 5.3.6.3 CONVERGENCE OF THE SCHEME 228 ..................... 5.3.6.4 STABILITY 228 ........... 5.4 SYMPLECTIC AND ENERGY-CONSERVING SCHEMES 229 5.4.1 CANONICAL HAMILTONIAN SYSTEMS ............. 230 ................ 5.4.1.1 SYMPLECTIC EULER 232 5.4.1.2 THE LINTRAP SCHEME .............. 234 5.4.2 NON-CANONICAL HAMILTONIAN SYSTEMS ........... 236 5.4.2.1 LINTRAP FOR LOTKA-VOLTERRA ........... 238 5.4.2.2 SYMPLECTIC EULER FOR LOTKA-VOLTERRA ...... 242 ......................... 5.5 ACKNOWLEDGEMENT 243 BIBLIOGRAPHY 244 APPENDIX A OTHER RELEVANT REFERENCES LIST OF CONTRIBUTORS
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spelling	Applications of nonstandard finite difference schemes ed.: Ronald E. Mickens Singapore [u.a.] World Scientific 2000 XII, 250 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential equations, Partial Numerical solutions Finite differences (DE-588)1071861417 Konferenzschrift gnd-content Mickens, Ronald E. Sonstige oth OEBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043962&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Applications of nonstandard finite difference schemes Differential equations, Partial Numerical solutions Finite differences
subject_GND	(DE-588)1071861417
title	Applications of nonstandard finite difference schemes
title_auth	Applications of nonstandard finite difference schemes
title_exact_search	Applications of nonstandard finite difference schemes
title_full	Applications of nonstandard finite difference schemes ed.: Ronald E. Mickens
title_fullStr	Applications of nonstandard finite difference schemes ed.: Ronald E. Mickens
title_full_unstemmed	Applications of nonstandard finite difference schemes ed.: Ronald E. Mickens
title_short	Applications of nonstandard finite difference schemes
title_sort	applications of nonstandard finite difference schemes
topic	Differential equations, Partial Numerical solutions Finite differences
topic_facet	Differential equations, Partial Numerical solutions Finite differences Konferenzschrift
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043962&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT mickensronalde applicationsofnonstandardfinitedifferenceschemes

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