Numerical Mathematics and Applications (eBook)

Front Cover 1
Numerical Mathematics and Applications 4
Copyright Page 5
Table of Contents 8
FOREWORD 6
Section I: ACCURACY OF COMPUTATION 14
CHAPTER 1. THE USE OF THE CESTAC METHOD IN THE PARALLEL COMPUTATION OF ROOTS OF POLYNOMIALS 16
I. THE CESTAC METHOD 16
II - THE PARALLEL COMPUTATION OF ROOTS OF POLYNOMIALS 17
Ill - THE INITIALIZATION AND TERMINATION OF THE ITERATIVE PROCESS 18
IV - THE ESTIMATION OF ACCURACY OF THE COMPUTED ROOTS 21
V - CONCLUSION 21
REFERENCES 21
CHAPTER 2. CESTAC, A TOOL FOR A STOCHASTIC ROUND-OFF ERROR ANALYSIS IN SCIENTIFIC COMPUTING 24
1. Introduction 24
2. The Arithmetic of Computers and the arising Problems 25
3. The Perturbation Method, Validity of a computed Result 25
4. The Implementation 26
5. CESTAC Tested on an Algorithm and Numerical Results 27
6. Conclusion 32
REFERENCES 33
CHAPTER 3. REDUCING ABBREVIATION ERRORS IN ITERATIVE RESOLUTION OF LINEAR SYSTEMS 34
1. INTRODUCTION 34
2. NOTATIONS AND THEORETICAL PROCESS 34
3. THE SEQUENCE OF ITERATES IN FLOATING-POINT ARITHMETIC 35
4. THE SET E={X : X € FN^ X=B R X + c} 36
5. A SUFFICIENT CONDITION FOR THE EXISTENCE OF VECTORS X SOLUTIONS IN SET FN 36
6. CONSEQUENCES 37
7. VALIDITY OF COMPUTED SOLUTION 37
REFERENCES 38
CHAPTER 4. OPTIMAL TERMINATION CRITERION AND ACCURACY TESTS IN MATHEMATICAL PROGRAMMING 40
1. INTRODUCTION 40
2. LINEAR PROGRAMMING 40
3. THE KARMARKAR ALGORITHM [ 3 ] 42
4. CONSTRAINED NON-LINEAR PROGRAMMING 44
5. REFERENCES 44
CHAPTER 5. ON THE USE OF THE NORMED RESIDUE TO CHECK THE QUALITY OF THE SOLUTION OF A LINEAR SYSTEM 46
ABSTRACT 46
0. INTRODUCTION 46
1. NOTATIONS AND HYPOTHESIS 46
2. ANALYSIS OF RELATIVE ERRORS 47
3 - NUMBER OF EXACT DIGITS 48
4. - INCIDENCE OF ROUNDING ERRORS 49
5. COMPARISON WITH THE NORMED RESIDUE 50
6 - EXAMPLE 50
7 - CONCLUSION 52
REFERENCES 52
CHAPTER 6. COMPUTABLE BOUNDS FOR SOLUTIONS OF INTEGRAL EQUATIONS 54
INTEGRAL EQUATIONS 54
INTERVAL ANALYSIS 54
INTERVAL INTEGRATION 55
INTERVAL ITERATION 55
INCLUSION OF INTEGRAL OPERATORS 55
MONOTONICITY METHODS 56
DIRECTED ROUNDING 56
ITERATIVE RESIDUAL CORRECTION 56
EIGENVALUE PROBLEMS 57
REFERENCES 58
CHAPTER 7. Arbitrarily Accurate Boundaries for Solutions of ODEs with Initial Values using Variable Precision Arithmetic 60
Abstract 60
1 Introduction 60
2 The Multi Level Structure 60
3 Elements of the Numerical Basis 61
4 The Initial Value Problem 62
5 The Algorithm 63
6 The Inclusion of the Solution 65
7 Conclusions 66
References 66
Section II: APPROXIMATIONS AND ALGORITHMS 68
CHAPTER 8. REMARKS ON SOME MODIFIED ROMBERG ALGORITHMS FOR NUMERICAL INTEGRATION 70
SUMMARY 70
1. INTRODUCTION 70
2. PROBLEM FORMULATION 70
3. EXTRAPOLATION SCHEMES 73
4. SOME REMARKS ON THE IMPLEMENTATION OF THE ALGORITHM 75
5. NUMERICAL EXPERIMENTS 76
REFERENCES 76
CHAPTER 9. LINEAR AND QUASILINEAR EXTRAPOLATION ALGORITHMS 78
0. INTRODUCTION, NOTATION AND DEFINITIONS 78
1. A GENERAL DETERMINANTAL IDENTITY 78
2. RECURRENCE FORMULAS FOR EXTRAPOLATION 79
3. PARTICULAR CASES AND APPLICATIONS 81
REFERENCES 83
CHAPTER 10. VECTOR PADÉ APPROXIMANTS 86
NATURE OF THE PROBLEM 86
I - PADÉ-TYPE APPROXIMANTS 86
II - IMPROVEMENT OF THE ORDER OF APPROXIMATION 87
Ill - DENOMINATORS OF THE VECTOR PADÉ APPROXIMANTS 87
IV- PADÉ APPROXIMANTS AS RATIO OF TWO DETERMINANTS 88
V - RECURSIVE COMPUTATION OF THE VECTOR PADÉ-APPROXIMANTS 89
REFERENCES 90
CHAPTER 11. THREE COMPUTATIONAL ASPECTS OF CONTINUED FRACTION/PADÉ APPROXIMANTS 92
1. INTRODUCTION 92
1. SPEED OF CONVERGENCE AND A PRIORI TRUNCATION ERROR ESTIMATES 94
2. ACCELERATION OF CONVERGENCE 95
3. ANALYTIC CONTINUATION AND NUMERICAL STABILITY 96
REFERENCES 97
CHAPTER 12. AN UNIVERSAL CONVERSATIONAL PROGRAM FOR COMPUTING SEQUENCES OF PADE APPROXIMANTS IN THE NON NORMAL CASE 98
SUMMARY 98
1. THE NORMAL CASE 98
2. THE NON NORMAL CASE 99
3. THE CONVERSATIONAL PROGRAM [5] 99
4. NUMERICAL EXAMPLE [6] 100
5 - CONCLUSION 100
BIBLIOGRAPHY 100
CHAPTER 13. EFFICIENT COMPUTATION OF A GROUP OF CLOSE EIGENVALUES FOR INTEGRAL OPERATORS 102
ABSTRACT 102
1. INTRODUCTION 102
2. THEORETICAL ASPECTS 102
3. PRACTICAL COMPUTATIONS 104
4. NUMERICAL EXAMPLES 105
5. FINAL REMARKS 105
REFERENCES 106
CHAPTER 14. Data Flow Analysis of Orthogonal Properties on the Conjugate Gradient and the Lanczos Algorithm 108
1. Introduction 108
2. The Conjugate Gradient and Lanczos Method 108
3. Propagation of Orthogonality 110
4. Numerical Experiments 111
Acknowledgment 112
References 112
Section III: SOLUTION OF ODE'S AND PDE'S 118
CHAPTER 15. EFFICIENT PRECONDITIONINGS FOR MATRIX PROBLEMS RESULTING FROM HIGH ORDER METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 120
1. INTRODUCTION 120
2. PRECONDITIONING TECHNIQUES 122
3. SOLUTION PROCEDURES 123
4. NUMERICAL RESULTS 124
5. CONCLUSIONS 127
ACKNOWLEDGEMENT 128
REFERENCES 128
CHAPTER 16. VARIABLE STEP SIZE / VARIABLE ORDER PDE SOLVER WITH GLOBAL OPTIMISATION 130
1. INTRODUCTION 130
2. ACCESS TO THE ERROR 130
3. COMPUTATIONAL AMOUNT 131
4. SELECTION OF THE ORDER 132
5. SELECTION OF THE LOCAL STEP SIZES 133
6. FURTHER REMARKS TO THE CONTROL 134
7. EXAMPLES 134
Acknowledgement 135
8. REFERENCES 135
CHAPTER 17. APPROXIMATE PRACTICAL STABILITY FOR NONLINEAR EVOLUTION PDES 138
1. INTRODUCTION 138
2. THE PROBLEM FROM CONTINUUM MECHANICS 139
3. THE SPECIAL CASE OF THE LINEAR PDES 141
4. ON FOURIER-POLYNOMIALS FOR THE APPROXIMATION OF v, e, AND s 142
5. APPLICATION OF THE MEAN VALUE INTERVAL METHOD 142
6. THE ENCLOSURE CONDITION 143
7. THE NONLINEAR SAMPLE PROBLEM 144
8. INVESTIGATION CONCERNING INDIVIDUAL FOURIER-MODES 144
9. NUMERICAL RESULTS FOR THE SYSTEM (7.4) 145
10. CONCLUSIONS 146
REFERENCES 147
CHAPTER 18. CURRENT METHODS FOR LARGE STIFF ODE SYSTEMS 148
1. INTRODUCTION 148
2. SOLUTION METHOD TYPES 149
3. RECENT DEVELOPMENTS 151
REFERENCES 155
CHAPTER 19. EXPONENTIAL-FITTED METHODS FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS 158
1. INTRODUCTION 158
2. THE PROBLEM OF STIFFNESS 158
3. EXPONENTIAL-BASED ALGORITHMS 159
4. LOCAL ERROR ESTIMATES 161
5. DISCUSSION AND CONCLUSIONS 163
REFERENCES 164
CHAPTER 20. LINEARIZED D-MAPPING FOR STIFF COMPUTATIONS 166
1. INTRODUCTION 166
2. D-MATRIX AND D-MAPPING 166
3. LINEARIZED D-MAPPING ANALYSIS 166
4. CONVEX-TYPE OPERATION 167
5. AN APPLICATION 167
6. DISCUSSIONS 168
REFERENCES 168
CHAPTER 21. ON THE USE OF NEWTON'S METHOD IN THE ADAPTIVE SOLUTION OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS 170
1. Introduction 170
2. Preliminaries 170
3. Coarse-Fine Grid Relationship 171
4. Estimation of the Newton Kantorovich Norms 172
5. Assessment of the Convergence Estimates 172
6. Numerical Results 173
References 174
CHAPTER 22. SPLINE APPROXIMATIONS IN NUMERICAL METHOD OF LINES SOLUTION OF FIRST-ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS 176
1. INTRODUCTION 176
2. TEST PROBLEM 177
3. SPLINE DIFFERENTIATOR 177
4. HYBRIDIZED SPLINE 179
5. ADAPTIVE GRID 179
6. HYBRIDIZED ADAPTIVE GRID 181
7. CONCLUSIONS 181
NOTATION 181
SUBSCRIPTS 183
REFERENCES 183
CHAPTER 23. SOME INSIGHTS INTO THE STABILITY OF DIFFERENCE APPROXIMATIONS FOR HYPERBOLIC INITIAL-BOUNDARY-VALUE PROBLEMS 184
1. Introduction 184
2. IBVP for a model hyperbolic equation 184
3. A prototype difference scheme for the model IBVP 185
4. Lax-Richtmyer stability of a discrete IBVP 185
5. Difficulties in proving Lax-Richtmyer stability 186
6. Normal-mode analysis (quarter-plane problems) 186
7. A conjecture on a test forLax-Richtmyer stability 187
References 189
CHAPTER 24. USE OF A DYNAMIC GRID ADAPTION IN THE ASWR-METHOD 190
1. SUMMARY 190
2. INTRODUCTION 190
3. THE ASWR-METHOD ON A NONUNIFORM GRID 191
4. PRACTICAL CONSIDERATIONS 192
5. EXAMPLE 193
6. CONCLUSION 193
REFERENCES 194
CHAPTER 25. THE SOLUTION OF AN ELLIPTIC P.D.E. WITH PERIODIC BOUNDARY CONDITIONS IN A RECTANGULAR REGION 198
1. INTRODUCTION 198
2. PROBLEM DEFINITION 198
3. THE SOLUTION OF CONSTANT TERM CYCLIC TRIDIAGONAL MATRIX SYSTEMS 199
4. THE CYCLIC BLOCK FACTORISATION METHOD 200
5. THE SPECTRAL RESOLUTION METHOD 203
6. NUMERICAL EXPERIMENTS 204
REFERENCES 205
CHAPTER 26. A MODIFIED GALERKIN SCHEME FOR ELLIPTIC EQUATIONS WITH NATURAL BOUNDARY CONDITIONS 206
1. INTRODUCTION 206
2. BOUNDARY VALUE PROBLEMS 206
3. GALERKIN APPROXIMATIONS 209
4. CONCLUDING REMARKS 209
REFERENCES 210
CHAPTER 27. NUMERICAL GRID GENERATION THROUGH SECOND ORDER DIFFERENTIAL-GEOMETRIC MODELS 212
SUMMARY 212
INTRODUCTION 212
BASIC ELLIPTIC MODELS 212
NUMERICAL RESULTS 214
ACKNOWLEDGEMENT 216
REFERENCES 216
CHAPTER 28. FACTORIZATION AND PATH INTEGRATION OF THE HELMHOLTZ EQUATION: NUMERICAL ALGORITHMS 218
1. INTRODUCTION 218
2. FACTORIZATION AND PATH INTEGRATION 218
3. COMPUTATIONAL ALGORITHM 219
4. NUMERICAL RESULTS 220
5. DISCUSSION 220
6. REFERENCES 224
CHAPTER 29. A GENERAL ERGUN EQUATION FOR A MJLTILAYERED POROUS MEDIUM 226
1. INTRODUCTION 226
2. SUMMARY OF EXISTING THEORY 226
3. THE ANISOTROPIC APPROACH 228
4. CONCLUSION 232
5. LIST OF SYMBOLS 232
REFERENCES 233
CHAPTER 30. ITERATIVE SOLUTIONS OF PROBLEMS WITH SHOCKS 234
SUMMARY 234
1. NEWTON'S METHOD 234
2. TWO LEVEL ITERATIVE METHODS OF GRADIENT TYPE 234
3. THE BURGERS EQUATION 237
4. THE TRANSONIC SMALL DISTURBANCE EQUATION 240
5. CONCLUSION 242
ACKNOWLEDGEMENTS 243
REFERENCES 243
CHAPTER 31. THE LUMPED MASS FINITE ELEMENT METHOD FOR PARABOLIC EQUATIONS 244
SUMMARY 244
References 246
CHAPTER 32. THE SOLUTION OF BURGERS' EQUATION BY BOUNDARY VALUE METHODS 248
Abstract 248
1. Introduction 248
2. The Boundary Value Procedure 248
3· Iterative Methods of Solution 248
4. Iterative Methods of Solution 249
5. The Hopscotch Formulation of the Boundary Value Technique 250
6. Numerical Experiments 251
7. Conclusions 251
References 252
Section IV: COMPUTATIONAL ACOUSTICS 254
CHAPTER 33. NUMERICAL MODELS FOR OCEAN ACOUSTIC MODES 256
1. INTRODUCTION 256
2. SUMMARY OF THE NUMERICAL METHOD FOR THE STANDARD NORMAL MODE PROBLEM (1.1) 257
3. RESULTS 258
REFERENCES 258
CHAPTER 34. THE RELATION OF THE PARABOLIC EQUATION METHOD TO THE ADIABATIC MODE APPROXIMATION 262
1. INTRODUCTION 262
2. BASIC PROPAGATION PROBLEM 263
3. COUPLED MODES 263
4. ADIABATIC APPROXIMATIONS 264
5. ADIABATIC EQUIVALENCES 265
6. NATURAL REFERENCE WAVENUMBER 266
7. CONSTANT REFERENCE WAVENUMBER 267
8. CONCLUDING REMARKS 267
ACKNOWLEDGMENTS 267
REFERENCES 267
CHAPTER 35. A SURVEY OF NUMERICAL METHODS FOR A NEW CLASS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ARISING IN NONSPHERICAL GEOMETRICAL OPTICS 270
1. INTRODUCTION: FORMULATION OF THE GENERAL EQUATIONS 270
2. BOUNDARY VALUE PROBLEMS AND SOLUTION METHODS 271
3. EXISTENCE THEORY AND QUESTIONS 274
REFERENCES 275
CHAPTER 36. WIDE ANGLE PARABOLIC APPROXIMATIONS IN UNDERWATER ACOUSTICS 278
1. INTRODUCTION 278
2. COMPARISON OF PROPAGATION ANGLES 279
3. A NEW WIDE ANGLE PARABOLIC APPROXIMATION 280
4. TIME-DEPENDENT ONE-WAY WAVE EQUATIONS 283
ACKNOWLEDGEMENT 284
REFERENCES 284
CHAPTER 37. THE APPLICATION OF THE BOUNDARY INTEGRAL ELEMENT METHOD TO THE PROBLEM OF SCATTERING OF SOUND WAVES BY AN ELASTIC WEDGE 286
SUMMARY 286
FORMULATION OF THE PROBLEM 286
THE METHOD OF SOLUTION 287
SOME RESULTS 290
CONCLUSIONS 292
REFERENCES 292
ACKNOWLEDGEMENTS 292
CHAPTER 38. A WAVE PROPAGATION COMPUTATION TECHNIQUE USING FUNCTION THEORETIC REPRESENTATION 294
1. INTRODUCTION 294
2. TRANSMUTATION FROM IDEALIZED TO PERTURBED 294
3. THE BOUNDARY CONDITION AT z=b IS PRESERVED 295
4. SOME BOUNDARY TYPE CONDITIONS FOR THE KERNEL 295
5. DERIVATION OF THE PARTIAL DIFFERENTIAL EQUATION AND CONDITIONS THAT THE KERNEL MUST SATISFY 296
6. THE OTHER BOUNDARY CONDITION IS NOT PRESERVED 296
7. EXISTENCE AND UNIQUENESS OF THE TRANSMUTATION 297
8. FINDING GREEN'S FUNCTION BY HANKEL TRANSFORM 297
9. A SPECIAL EXAMPLE OF THE TRANSMUTATION KERNEL 298
10. FINDING KERNEL APPROXIMATIONS USING MACSYMA 299
ACKNOWLEDGEMENT 301
FOOTNOTES 301
REFERENCES 301
CHAPTER 39. COMPENSATING FOR WAVEFRONT TURNING IN WAVEFRONT CURVATURE RANGING 302
1. INTRODUCTION 302
2. CONVENTIONAL TRIANGULATION 302
3. WAVEFRONT TURNING "AT ENDFIRE" 303
4. WHAT CAN BE ACHIEVED AT ENDFIRE 304
5. MEASURING WAVEFRONT CURVATURE AT ENDFIRE 304
6. "NORMAL" RADIUS OF CURVATURE APPROXIMATES RANGE 305
7. CONCLUSIONS 305
ACKNOWLEDGMENT 306
REFERENCES 306
CHAPTER 40. CHANGES IN EIGENVALUES DUE TO BOTTOM INTERACTION USING PERTURBATION THEORY 308
1. INTRODUCTION 308
2. FORMULATION 308
3. PERTURBATION 309
4. EXAMPLES 311
ACKNOWLEDGEMENT 313
REFERENCES 314
Section V: COMPUTATIONAL FLUID DYNAMICS 316
CHAPTER 41. COMPUTATIONAL FLUID DYNAMICS, CONVERGENT OR ASYMPTOTIC 318
1. INTRODUCTION 318
2. NONLINEAR EQUIVALENCE THEOREM 319
3. ERROR ANALYSIS 320
4. ASYMPTOTICS 321
5. CONCLUDING REMARKS 322
ACKNOWLEDGEMENTS 322
REFERENCES 322
CHAPTER 42. HIGHLY ACCURATE SHOCK FLOW CALCULATIONS WITH MOVING GRIDS AND MESH REFINEMENT 324
1. INTRODUCTION 324
References 328
CHAPTER 43. MODIFIED EQUATION METHODS FOR ONE-DIMENSIONAL FLAME PROPAGATION PROBLEMS 330
1. INTRODUCTION 330
2. PROBLEM FORMULATION 331
3. MODIFIED EQUATION METHODS 332
4. STABILITY OF MODIFIED EQUATION METHODS 333
5. PRESENTATION OF RESULTS 335
6. CONCLUSIONS 338
REFERENCES 339
CHAPTER 44. NUMERICAL SOLUTION OF TIME-DEPENDENT INCOMPRESSIBLE FLOWS 340
1. INTRODUCTION 340
2. GOVERNING EQUATIONS AND DISCRETIZATION 340
3. THE NUMERICAL PROCEDURE 341
4. TEST PROBLEMS 341
5. CONCLUDING REMARKS 345
REFERENCES 345
CHAPTER 45. PSEUDOCHARACTERISTIC METHOD OF LINES SIMULATION OF SINGLE- AND TWO-PHASE ONE-DIMENSIONAL FLOW TRANSIENTS 346
1. INTRODUCTION 346
2. SINGLE-PHASE PROBLEM 347
3. TWO-PHASE PROBLEM 348
4. DISCUSSION 349
ACKNOWLEDGMENTS 349
REFERENCES 350
NOMENCLATURE 350
CHAPTER 46. LAGRANGIAN MODELING OF TURBULENT DISPERSION IN SHEAR LAYERS 352
1. INTRODUCTION 352
2. THE LAGRANGIAN APPROACH TO TURBULENT DISPERSION 352
3. IMPLEMENTATION AND RESULTS 354
4. EXTENSION TO BUOYANT CONTAMINANTS 357
5. CONCLUSION 358
ACKNOWLEDGEMENTS 358
REFERENCES 358
CHAPTER 47. STUDIES IN A SHALLOW WATER FLUID MODEL WITH TOPOGRAPHY 360
1. INTRODUCTION 360
2. NUMERICAL MODEL 360
3. NUMERICAL SIMULATIONS 361
4. PHASE SPEED 362
REFERENCES 364
APPENDIX 364
CHAPTER 48. COMPUTATION OF THE FINE VORTEX STRUCTURES OF FLUIDS 368
1. INTRODUCTION 368
2. FLUID FINE VORTEX STRUCTURE RESULTS 370
3. ROBUST COMPUTATION OF FLUID FINE STRUCTURE 373
4. ADDITIONAL REMARKS 374
ACKNOWLEDGEMENTS 375
REFERENCES 375
CHAPTER 49. INVISCID VORTEX FLOW SIMULATIONS BY MEGACELL SOLUTIONS TO THE EULER EQUATIONS 378
Summary 378
Introduction 378
Numerical Solution Procedure 378
Simulated Supersonic Vortex Flowfields 379
References 379
CHAPTER 50. TRANSONIC POTENTIAL FLOWS: IMPROVED ACCURACY BY USING LOCAL GRIDS 382
SUMMARY 382
1. INTRODUCTION 382
2. GOVERNING EQUATIONS AND FINITE-DIFFERENCE APPROXIMATION 382
3. LOCAL MESH REFINEMENTS 383
4. NUMERICAL RESULTS 384
REFERENCES 388
CHAPTER 51. NUMERICAL ANALYSIS OF UNSTEADY WAKE DEVELOPMENT BEHIND AN IMPULSIVELY STARTED CYLINDER IN SLIGHTLY VISCOUS FLUID 390
Abstract 390
1. Introduction 390
2. The Numerical Method in 2-D 390
3. Numerical Parameters 392
4. Evolution of the Vorticity Peak 393
5. Development of 'Vortex' Structures in the Recirculating Zones 393
6. Calculated Numerical Functionals 396
7. Conclusion 398
References 398
CHAPTER 52. NUMERICAL TREATMENT OF SHOCKS IN UNSTEADY POTENTIAL FLOW COMPUTATION 400
ABSTRACT 400
1. INTRODUCTION 400
2. ONE-DIMENSIONAL SHOCK WAVE MOTION 400
3. TWO-DIMENSIONAL UNSTEADY TRANSONIC POTENTIAL FLOW 403
REFERENCES 405
CHAPTER 53. GENERALIZED VORTEX METHODS FOR STRATIFIED LAYERED FLOWS 406
1. STRATIFIED VORTEX FLOWS 406
2. INTERFACIAL FLOW 406
3. NUMERICAL TECHNIQUES 407
4. APPLICATIONS 407
5. COMMENTS AND CONCLUSIONS 409
REFERENCES 409
CHAPTER 54. A TWO-FLUID MODEL OF TURBULENCE APPLIED TO SIMULATION OF FIRES 412
Abstract 412
Introduction 412
The Physical Process Considered 413
The Simulated Physical Process 413
The Differential Equations Conservation of Mass 413
Conservation of a General Dependent Variable 413
The Interfluid Relations Interfluid Mass Transfer 413
Interfluid Friction 414
The Modelling of Turbulence 414
The Combustion Model 415
The Second-Fluid Density 415
Parametric Studies 415
Results 415
Computer Requirements 418
Conclusions 418
References 418
CHAPTER 55. NUMERICAL PREDICTION OF TURBULENT FLOW OVER A SURFACE-MOUNTED CUBE 420
Summary 420
Introduction 420
Experiment 420
Mathematical Formulation 421
Method of Solution 422
Computational Detail 423
Presentation and Discussion of Results 423
Conclusions 424
References 424
CHAPTER 56. NUMERICAL MODELLING OF AIR FLOW IN CONFINED TAPERED DUCT INLETS 428
1. INTRODUCTION 428
2. MATHEMATICAL MODELLING 428
3. NUMERICAL PROCEDURE 428
4. BOUNDARY CONDITIONS 429
5. COMPUTATION OF THE POTENTIAL FLOW FIELD 429
6. NUMERICAL RESULTS 430
7. DISCUSSION 432
REFERENCES 433
CHAPTER 57. SUBSURFACE FLUID DYNAMICS AND TRANSPORT PHENOMENA BASED ON A VECTOR REPRESENTATION 434
1. INTRODUCTION 434
2. THEORETICAL FOUNDATIONS 434
3. NUMERICAL METHODS 437
4. THE COMPUTER PROGRAM 439
REFERENCES 441
AUTHOR INDEX 442