Solving PDEs in C++: numerical methods in a unified object-oriented approach
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
2006
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| Series: | Computational science and engineering
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXIII, 500 S. graph. Darst. |
| ISBN: | 0898716012 9780898716016 |
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MARC
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| 100 | 1 | |a Shapira, Yair |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Solving PDEs in C++ |b numerical methods in a unified object-oriented approach |c Yair Shapira |
| 264 | 1 | |a Philadelphia |b Society for Industrial and Applied Mathematics |c 2006 | |
| 300 | |a XXIII, 500 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Computational science and engineering | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a C++ (linguagem de programação) |2 larpcal | |
| 650 | 7 | |a Equações diferenciais parciais |2 larpcal | |
| 650 | 7 | |a Programação orientada a objetos |2 larpcal | |
| 650 | 4 | |a C++ (Computer program language) | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Object-oriented programming (Computer science) | |
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| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a C++ |0 (DE-588)4193909-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
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| adam_text | SOLVING PDES IN C++ NUMERICAL METHODS IN A UNIFIED OBJECT-ORIENTED
APPROACH YAIR SHAPIRA TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY HAIFA,
ISRAEL SIAM. SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA
CONTENTS LIST OF FIGURES XIII LIST OF TABLES XXI PREFACE XXIII I
PROGRAMMING 1 1 INTRODUCTION TO C 7 1.1 VARIABLES AND TYPES 7 1.2
ASSIGNMENT 8 1.3 INITIALIZATION 8 1.4 CONVERSION 9 1.5 ARITHMETIC
OPERATIONS 9 1.6 FUNCTIONS 11 1.7 PRINTING OUTPUT 13 1.8 CONDITIONS 14
1.9 SCOPE OF VARIABLES 15 1.10 LOOPS 17 1.11 EXAMPLES WITH LOOPS 19 1.12
EXAMPLE: REVERSED INTEGER 20 1.13 POINTERS 22 1.14 ARRAYS 23 1.15
PASSING ARGUMENTS TO FUNCTIONS 24 1.16 I/O 25 1.17 RECURSION 26 1.18
EXAMPLE: BINARY REPRESENTATION 27 1.19 EXAMPLE: PASCAL S TRIANGLE 29
1.20 EXAMPLE: LOCAL MAXIMUM 30 1.21 EXAMPLE: ARITHMETIC EXPRESSION 36
1.22 EXAMPLE: THE EXPONENT FUNCTION 40 1.23 EXERCISES 43 VI CONTENTS 2
INTRODUCTION TO C+ + 47 2.1 OBJECTS 47 2.2 CLASSES 48 2.3 CONSTRUCTORS
51 2.4 EXPLICIT CONVERSION 53 2.5 IMPLICIT CONVERSION 53 2.6 THE DEFAULT
COPY CONSTRUCTOR 53 2.7 DESTRUCTOR 55 2.8 MEMBER AND FRIEND FUNCTIONS 55
2.9 REFERENCES 57 2.10 COPY CONSTRUCTOR 59 2.11 ASSIGNMENT OPERATORS 60
2.12 OPERATORS 62 2.13 INVERSE CONVERSION 63 2.14 UNARY OPERATORS 63
2.15 BINARY OPERATORS 65 2.16 EXAMPLE: COMPLEX NUMBERS 68 2.17 TEMPLATES
72 2.18 EXAMPLE: THE VECTOR OBJECT 75 2.19 INHERITANCE 79 2.20 EXAMPLE:
THE MATRIX OBJECT 86 2.21 DETERMINANT AND INVERSE OF A SQUARE MATRIX 88
2.22 EXPONENT OF A SQUARE MATRIX 90 2.23 EXERCISES 90 3 DATA STRUCTURES
93 3.1 DATA STRUCTURES 93 3.2 TEMPLATES IN DATA STRUCTURES 94 3.3
DYNAMIC VECTORS 94 3.4 LISTS 99 3.5 CONNECTED LISTS 102 3.6 THE MERGING
PROBLEM 108 3.7 THE ORDERING PROBLEM 110 3.8 VECTORS VS. LISTS ILL 3.9
TWO-SIDED CONNECTED LISTS 112 3.10 TREES 112 3.11 GRAPHS 114 3.12
EXERCISES 115 II THE OBJECT-ORIENTED APPROACH 117 4 OBJECT-ORIENTED
PROGRAMMING 121 4.1 OBJECT-ORIENTED LANGUAGE 121 4.2 EXAMPLE: THE
GRAPH-COLORING PROBLEM 122 4.3 DOWNWARD IMPLEMENTATION 124 CONTENTS VII
4.4 THE C++IMPLEMENTATION 126 4.5 TRIANGULATION 128 4.6 EXAMPLE: THE
TRIANGLE-COLORING PROBLEM 129 4.7 DOWNWARD IMPLEMENTATION 130 4.8
SEPARATION OF INFORMATION 131 4.9 APPLICATION IN NUMERICAL SCHEMES 133
4.10 EXERCISES 134 5 ALGORITHMS AND THEIR OBJECT-ORIENTED IMPLEMENTATION
135 5.1 IDEAS AND THEIR IMPLEMENTATION . . . 135 5.2 MULTILEVEL
PROGRAMMING 136 5.3 INFORMATION AND STORAGE 137 5.4 EXAMPLE: THE
POLYNOMIAL OBJECT 138 5.5 MULTIPLICATION OF POLYNOMIALS 141 5.6
CALCULATION OF A POLYNOMIAL 143 5.7 ALGORITHMS AND THEIR IMPLEMENTATION
143 5.8 HORNER S ALGORITHM 144 5.9 CALCULATION OF A POWER 145 5.10
CALCULATION OF DERIVATIVES 147 5.11 THE TAYLOR EXPANSION 148 5.12
DERIVATIVES OF A PRODUCT 151 5.13 POLYNOMIAL OF TWO VARIABLES 152 5.14
INTEGRATION OF A POLYNOMIAL 154 5.15 EXERCISES 156 6 OBJECT-ORIENTED
ANALYSIS 161 6.1 ODES ; 161 6.2 STABILITY IN THE ODE 162 6.3 SYSTEM OF
ODES 162 6.4 STABILITY IN A SYSTEM OF ODES 163 6.5 STABLE INVARIANT
SUBSPACE 164 6.6 THE INHOMOGENEOUS CASE 165 6.7 NUMERICAL SOLUTION 165
6.8 DIFFERENCE SCHEMES 166 6.9 THE TAYLOR DIFFERENCE SCHEME 167 6.10
COMPUTATIONAL ERROR ESTIMATES 168 6.11 NONLINEAR ODES 170 6.12
OBJECT-ORIENTED ANALYSIS 170 6.13 APPLICATION 171 6.14 TAYLOR SCHEME
WITH ERROR ESTIMATES 172 6.15 ASYMPTOTIC SOLUTION 174 6.16 EXERCISES 176
VIII CONTENTS III PARTIAL DIFFERENTIAL EQUATIONS AND THEIR
DISCRETIZATION 179 7 THE CONVECTION-DIFFUSION EQUATION 187 7.1
INITIAL-BOUNDARY-VALUE PROBLEMS 187 7.2 FINITE DIFFERENCES 188 7.3 THE
UPWIND SCHEME 189 7.4 DISCRETE BOUNDARY CONDITIONS 190 7.5 THE EXPLICIT
SCHEME 191 7.6 THE IMPLICIT SCHEME 193 7.7 THE SEMI-IMPLICIT SCHEME 193
7.8 THE IMPLEMENTATION 194 7.9 HIERARCHY OF OBJECTS 198 7.10 LIST OF
VECTORS 198 7.11 THE TIME-SPACE GRID 199 7.12 DIFFERENCE OPERATORS 201
7.13 TWO SPATIAL DIMENSIONS 205 7.14 EXERCISES 207 8 STABILITY ANALYSIS
209 8.1 PRELIMINARIES 209 8.2 ALGEBRAIC REPRESENTATION 211 8.3 STABILITY
IN TIME MARCHING 212 8.4 ACCURACY AND ADEQUACY 214 8.5 EXERCISES 216 9
NONLINEAR EQUATIONS 219 9.1 NONLINEAR PDES 219 9.2 THE RIEMANN PROBLEM
219 9.3 CONFLICTING CHARACTERISTIC LINES 220 9.4 THE GODUNOV SCHEME 222
9.5 THE RANDOM-CHOICE SCHEME 225 9.6 THE N-WAVE 226 9.7 SINGULAR
PERTURBATION 226 9.8 LINEARIZATION 228 9.9 ADEQUACY IN THE LINEARIZED
PROBLEM . 230 9.10 THE INHOMOGENEOUS CASE 232 9.11 SYSTEM OF NONLINEAR
PDES 233 9.12 EXERCISES 235 10 APPLICATION IN IMAGE PROCESSING 237 10.1
DIGITAL IMAGES 237 10.2 THE DENOISING PROBLEM 237 10.3 THE NONLINEAR
DIFFUSION PROBLEM 238 10.4 THE DISCRETIZATION 239 10.5 LINEARIZATION 240
10.6 COLOR IMAGES 240 CONTENTS IX 10.7 DENOISING COLOR IMAGES 241 10.8
NUMERICAL EXAMPLES 242 10.9 EXERCISES 243 IV THE FINITE-ELEMENT
DISCRETIZATION METHOD 245 11 THE WEAK FORMULATION 249 11.1 THE DIFFUSION
PROBLEM 249 11.2 THE WEAK FORMULATION 250 11.3 THE MINIMIZATION PROBLEM
252 11.4 THE COERCIVITY PROPERTY 253 11.5 EXISTENCE THEOREM 254 11.6
UNIQUENESS THEOREM 255 11.7 EXERCISES 256 12 LINEAR FINITE ELEMENTS 259
12.1 THE FINITE-ELEMENT TRIANGULATION 259 12.2 THE DISCRETE WEAK
FORMULATION 260 12.3 THE STIFFNESS SYSTEM 260 12.4 PROPERTIES OF THE
STIFFNESS MATRIX 261 12.5 CALCULATING THE STIFFNESS MATRIX 263 12.6
EXAMPLE: RECTANGULAR DOMAIN AND UNIFORM MESH 266 12.7 M-MATRIX PROPERTY
IN THE ISOTROPIC CASE 269 12.8 HIGHLY ANISOTROPIC EQUATIONS 271 12.9
EXAMPLE: CIRCULAR DOMAIN 274 12.10 EXERCISES 276 12.11 ADVANCED
EXERCISES 277 13 UNSTRUCTURED FINITE-ELEMENT MESHES 281 13.1 CONCRETE
AND ABSTRACT OBJECTS 281 13.2 THE NODE OBJECT 284 13.3 THE
FINITE-ELEMENT OBJECT 286 13.4 THE MESH OBJECT 291 13.5 ASSEMBLING THE
STIFFNESS MATRIX 293 13.6 EXERCISES 296 14 ADAPTIVE MESH REFINEMENT 297
14.1 LOCAL REFINEMENT 297 14.2 ADAPTIVE REFINEMENT 299 14.3 THE
ADAPTIVE-REFINEMENT ALGORITHM 299 14.4 PRESERVING CONFORMITY 301 14.5
PRESERVING CONFORMITY IN PRACTICE 302 14.6 MESH REFINEMENT IN PRACTICE
304 14.7 AUTOMATIC BOUNDARY REFINEMENT 307 14.8 IMPLEMENTATION OF
AUTOMATIC BOUNDARY REFINEMENT 307 X CONTENTS 14.9 NONCONVEX DOMAINS 309
14.10 EXERCISES 312 15 HIGH-ORDER FINITE ELEMENTS 313 15.1 HIGH-ORDER
VS. LINEAR FINITE ELEMENTS 313 15.2 QUADRATIC FINITE ELEMENTS 314 15.3
CUBIC FINITE ELEMENTS 316 15.4 PROS AND CONS 317 15.5 EXERCISES 318 V
THE NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS OF EQUATIONS 321
16 SPARSE MATRICES AND THEIR IMPLEMENTATION 325 16.1 SPARSE VS. DENSE
MATRICES 325 16.2 THE MATRIX-ELEMENT OBJECT 326 16.3 THE ROW OBJECT 329
16.4 THE SPARSE-MATRIX OBJECT 334 16.5 ASSEMBLING THE SPARSE STIFFNESS
MATRIX 337 16.6 EXERCISES 338 17 ITERATIVE METHODS FOR LARGE SPARSE
LINEAR SYSTEMS 341 17.1 ITERATIVE VS. DIRECT METHODS 341 17.2 ITERATIVE
METHODS 342 17.3 GAUSS-SEIDEL RELAXATION 344 17.4 JACOBI RELAXATION 344
17.5 SYMMETRIC GAUSS-SEIDEL 346 17.6 THE NORMAL EQUATION 346 17.7
INCOMPLETE FACTORIZATION 347 17.8 THE MULTIGRID METHOD 351 17.9
ALGEBRAIC MULTIGRID (AMG) 354 17.10 IMPLEMENTATION OF MULTIGRID 355
17.11 PRECONDITIONED CONJUGATE GRADIENTS (PCGS) 359 17.12 MULTIGRID FOR
NONSYMMETRIC LINEAR SYSTEMS 362 17.13 DOMAIN DECOMPOSITION AND MULTIGRID
362 17.14 EXERCISES 366 18 PARALLELISM 369 18.1 PARALLEL VS. SEQUENTIAL
COMPUTERS 369 18.2 PARALLELIZABLE ALGORITHMS 370 18.3 INHERENTLY
SEQUENTIAL ALGORITHMS 370 18.4 ARCHITECTURE 370 18.5 CACHE-ORIENTED
RELAXATION 371 18.6 SCHWARZ BLOCK RELAXATION 373 18.7 PARALLEL
ARCHITECTURES 373 18.8 SHARED MEMORY 373 CONTENTS XI 18.9 DISTRIBUTED
MEMORY 374 18.10 COMMUNICATION NETWORK 375 18.11 HYPERCUBE 376 18.12
EXAMPLE: MULTIPLICATION OF SPARSE MATRICES 378 18.13 LOW-LEVEL
C++IMPLEMENTATION 379 18.14 COMPARING ARCHITECTURES 380 18.15 JACOBI
RELAXATION 382 18.16 CONVERGENCE ANALYSIS 383 18.17 EXAMPLES FROM
HARWELL-BOEING 385 18.18 BLOCK JACOBI RELAXATION 388 18.19 EXERCISES 389
VI APPLICATIONS 391 19 DIFFUSION EQUATIONS 395 19.1 THE BOUNDARY-VALUE
PROBLEM 395 19.2 THE FINITE-ELEMENT MESH 396 19.3 THE LINEAR-SYSTEM
SOLVER 398 19.4 IMPLICIT TIME MARCHING 399 19.5 EXERCISES 401 20 THE
LINEAR ELASTICITY EQUATIONS 403 20.1 SYSTEM OF PDES 403 20.2 THE STRONG
FORMULATION 403 20.3 THE WEAK FORMULATION 404 20.4 THE MINIMIZATION
PROBLEM 405 20.5 COERCIVITY OF THE QUADRATIC FORM 405 20.6 THE
FINITE-ELEMENT DISCRETIZATION . 406 20.7 THE STIFFNESS MATRIX 407 20.8
THE ADAPTIVE-REFINEMENT CRITERION 408 20.9 THE MODIFIED MULTIGRID
ALGORITHM 408 20.10 NUMERICAL EXAMPLES 409 20.11 EXERCISES 412 21 THE
STOKES EQUATIONS 413 21.1 THE NABLA OPERATOR 413 21.2 GENERAL LINEAR
ELASTICITY 416 21.3 REDUCTION TO THE LINEAR ELASTICITY EQUATIONS 417
21.4 THE STOKES EQUATIONS 418 21.5 CONTINUATION PROCESS 418 21.6
ADEQUACY CONSIDERATION 419 21.7 PRECONDITIONER FOR THE STOKES EQUATIONS
419 21.8 FIRST-ORDER SYSTEM LEAST SQUARES 422 21.9 THE NAVIER-STOKES
EQUATIONS 423 21.10 EXERCISES 424 XII CONTENTS 22 ELECTROMAGNETIC WAVES
425 22.1 THE WAVE EQUATION 425 22.2 THE HELMHOLTZ EQUATION 426 22.3
FINITE-DIFFERENCE DISCRETIZATION 427 22.4 ADEQUACY IN FINITE DIFFERENCES
427 22.5 BILINEAR FINITE ELEMENTS 428 22.6 ADEQUACY IN BILINEAR FINITE
ELEMENTS 429 22.7 THE MEASURING PROBLEM 430 22.8 THE NEARLY RECTANGULAR
MODEL 431 22.9 THE NEARLY CIRCULAR MODEL 432 22.10 NONLINEAR FINITE
ELEMENTS 433 22.11 THE MAXWELL EQUATIONS 434 22.12 REDUCTION TO
THREE-DIMENSIONAL HELMHOLTZ EQUATIONS 436 22.13 EXERCISES 437 APPENDIX
439 A. 1 OPERATION S WITH VECTORS 439 A.2 OPERATIONS WITH MATRICES 440
A.3 OPERATIONS WITH DYNAMIC VECTORS 442 A.4 TWO-DIMENSIONAL
CONVECTION-DIFFUSION EQUATION 444 A.5 STABILITY IN THE EXPLICIT SCHEME
456 A.6 JORDAN FORM OF A TRIDIAGONAL MATRIX 456 A.7 DENOISING DIGITAL
IMAGES 458 A.8 MEMBERS OF THE MESH CLASS 460 A.9 OPERATIONS WITH SPARSE
MATRICES 462 A.10 KACMARZ ITERATION 465 A.LL ILU ITERATION 465 A. 12
MULTIGRID ITERATION 466 A. 13 ACCELERATION TECHNIQUES 470 A.14 PARALLEL
IMPLEMENTATION 475 A. 15 THE ADAPTIVE-REFINEMENT CODE 476 A.16 THE
DIFFUSION SOLVER 478 A.17 THE LINEAR ELASTICITY SOLVER 481 BIBLIOGRAPHY
485 INDEX 489
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| adam_txt |
SOLVING PDES IN C++ NUMERICAL METHODS IN A UNIFIED OBJECT-ORIENTED
APPROACH YAIR SHAPIRA TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY HAIFA,
ISRAEL SIAM. SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA
CONTENTS LIST OF FIGURES XIII LIST OF TABLES XXI PREFACE XXIII I
PROGRAMMING 1 1 INTRODUCTION TO C 7 1.1 VARIABLES AND TYPES 7 1.2
ASSIGNMENT 8 1.3 INITIALIZATION 8 1.4 CONVERSION 9 1.5 ARITHMETIC
OPERATIONS 9 1.6 FUNCTIONS 11 1.7 PRINTING OUTPUT 13 1.8 CONDITIONS 14
1.9 SCOPE OF VARIABLES 15 1.10 LOOPS 17 1.11 EXAMPLES WITH LOOPS 19 1.12
EXAMPLE: REVERSED INTEGER 20 1.13 POINTERS 22 1.14 ARRAYS 23 1.15
PASSING ARGUMENTS TO FUNCTIONS 24 1.16 I/O 25 1.17 RECURSION 26 1.18
EXAMPLE: BINARY REPRESENTATION 27 1.19 EXAMPLE: PASCAL'S TRIANGLE 29
1.20 EXAMPLE: LOCAL MAXIMUM 30 1.21 EXAMPLE: ARITHMETIC EXPRESSION 36
1.22 EXAMPLE: THE EXPONENT FUNCTION 40 1.23 EXERCISES 43 VI CONTENTS 2
INTRODUCTION TO C+ + 47 2.1 OBJECTS 47 2.2 CLASSES ' 48 2.3 CONSTRUCTORS
51 2.4 EXPLICIT CONVERSION 53 2.5 IMPLICIT CONVERSION 53 2.6 THE DEFAULT
COPY CONSTRUCTOR 53 2.7 DESTRUCTOR 55 2.8 MEMBER AND FRIEND FUNCTIONS 55
2.9 REFERENCES 57 2.10 COPY CONSTRUCTOR 59 2.11 ASSIGNMENT OPERATORS 60
2.12 OPERATORS 62 2.13 INVERSE CONVERSION 63 2.14 UNARY OPERATORS 63
2.15 BINARY OPERATORS 65 2.16 EXAMPLE: COMPLEX NUMBERS 68 2.17 TEMPLATES
72 2.18 EXAMPLE: THE VECTOR OBJECT 75 2.19 INHERITANCE 79 2.20 EXAMPLE:
THE MATRIX OBJECT 86 2.21 DETERMINANT AND INVERSE OF A SQUARE MATRIX 88
2.22 EXPONENT OF A SQUARE MATRIX 90 2.23 EXERCISES 90 3 DATA STRUCTURES
93 3.1 DATA STRUCTURES 93 3.2 TEMPLATES IN DATA STRUCTURES 94 3.3
DYNAMIC VECTORS 94 3.4 LISTS 99 3.5 CONNECTED LISTS 102 3.6 THE MERGING
PROBLEM 108 3.7 THE ORDERING PROBLEM 110 3.8 VECTORS VS. LISTS ILL 3.9
TWO-SIDED CONNECTED LISTS 112 3.10 TREES 112 3.11 GRAPHS 114 3.12
EXERCISES 115 II THE OBJECT-ORIENTED APPROACH 117 4 OBJECT-ORIENTED
PROGRAMMING 121 4.1 OBJECT-ORIENTED LANGUAGE 121 4.2 EXAMPLE: THE
GRAPH-COLORING PROBLEM 122 4.3 DOWNWARD IMPLEMENTATION 124 CONTENTS VII
4.4 THE C++IMPLEMENTATION 126 4.5 TRIANGULATION 128 4.6 EXAMPLE: THE
TRIANGLE-COLORING PROBLEM 129 4.7 DOWNWARD IMPLEMENTATION 130 4.8
SEPARATION OF INFORMATION 131 4.9 APPLICATION IN NUMERICAL SCHEMES 133
4.10 EXERCISES 134 5 ALGORITHMS AND THEIR OBJECT-ORIENTED IMPLEMENTATION
135 5.1 IDEAS AND THEIR IMPLEMENTATION . . . 135 5.2 MULTILEVEL
PROGRAMMING 136 5.3 INFORMATION AND STORAGE 137 5.4 EXAMPLE: THE
POLYNOMIAL OBJECT 138 5.5 MULTIPLICATION OF POLYNOMIALS 141 5.6
CALCULATION OF A POLYNOMIAL 143 5.7 ALGORITHMS AND THEIR IMPLEMENTATION
143 5.8 HORNER'S ALGORITHM 144 5.9 CALCULATION OF A POWER 145 5.10
CALCULATION OF DERIVATIVES 147 5.11 THE TAYLOR EXPANSION 148 5.12
DERIVATIVES OF A PRODUCT 151 5.13 POLYNOMIAL OF TWO VARIABLES 152 5.14
INTEGRATION OF A POLYNOMIAL 154 5.15 EXERCISES 156 6 OBJECT-ORIENTED
ANALYSIS 161 6.1 ODES ; 161 6.2 STABILITY IN THE ODE 162 6.3 SYSTEM OF
ODES 162 6.4 STABILITY IN A SYSTEM OF ODES 163 6.5 STABLE INVARIANT
SUBSPACE 164 6.6 THE INHOMOGENEOUS CASE 165 6.7 NUMERICAL SOLUTION 165
6.8 DIFFERENCE SCHEMES 166 6.9 THE TAYLOR DIFFERENCE SCHEME 167 6.10
COMPUTATIONAL ERROR ESTIMATES 168 6.11 NONLINEAR ODES 170 6.12
OBJECT-ORIENTED ANALYSIS 170 6.13 APPLICATION 171 6.14 TAYLOR SCHEME
WITH ERROR ESTIMATES 172 6.15 ASYMPTOTIC SOLUTION 174 6.16 EXERCISES 176
VIII CONTENTS III PARTIAL DIFFERENTIAL EQUATIONS AND THEIR
DISCRETIZATION 179 7 THE CONVECTION-DIFFUSION EQUATION 187 7.1
INITIAL-BOUNDARY-VALUE PROBLEMS 187 7.2 FINITE DIFFERENCES 188 7.3 THE
UPWIND SCHEME 189 7.4 DISCRETE BOUNDARY CONDITIONS 190 7.5 THE EXPLICIT
SCHEME 191 7.6 THE IMPLICIT SCHEME 193 7.7 THE SEMI-IMPLICIT SCHEME 193
7.8 THE IMPLEMENTATION 194 7.9 HIERARCHY OF OBJECTS 198 7.10 LIST OF
VECTORS 198 7.11 THE TIME-SPACE GRID 199 7.12 DIFFERENCE OPERATORS 201
7.13 TWO SPATIAL DIMENSIONS 205 7.14 EXERCISES 207 8 STABILITY ANALYSIS
209 8.1 PRELIMINARIES 209 8.2 ALGEBRAIC REPRESENTATION 211 8.3 STABILITY
IN TIME MARCHING 212 8.4 ACCURACY AND ADEQUACY 214 8.5 EXERCISES 216 9
NONLINEAR EQUATIONS 219 9.1 NONLINEAR PDES 219 9.2 THE RIEMANN PROBLEM
219 9.3 CONFLICTING CHARACTERISTIC LINES 220 9.4 THE GODUNOV SCHEME 222
9.5 THE RANDOM-CHOICE SCHEME 225 9.6 THE N-WAVE 226 9.7 SINGULAR
PERTURBATION 226 9.8 LINEARIZATION 228 9.9 ADEQUACY IN THE LINEARIZED
PROBLEM . 230 9.10 THE INHOMOGENEOUS CASE 232 9.11 SYSTEM OF NONLINEAR
PDES 233 9.12 EXERCISES 235 10 APPLICATION IN IMAGE PROCESSING 237 10.1
DIGITAL IMAGES 237 10.2 THE DENOISING PROBLEM 237 10.3 THE NONLINEAR
DIFFUSION PROBLEM 238 10.4 THE DISCRETIZATION 239 10.5 LINEARIZATION 240
10.6 COLOR IMAGES 240 CONTENTS IX 10.7 DENOISING COLOR IMAGES 241 10.8
NUMERICAL EXAMPLES 242 10.9 EXERCISES 243 IV THE FINITE-ELEMENT
DISCRETIZATION METHOD 245 11 THE WEAK FORMULATION 249 11.1 THE DIFFUSION
PROBLEM 249 11.2 THE WEAK FORMULATION 250 11.3 THE MINIMIZATION PROBLEM
252 11.4 THE COERCIVITY PROPERTY 253 11.5 EXISTENCE THEOREM 254 11.6
UNIQUENESS THEOREM 255 11.7 EXERCISES 256 12 LINEAR FINITE ELEMENTS 259
12.1 THE FINITE-ELEMENT TRIANGULATION 259 12.2 THE DISCRETE WEAK
FORMULATION 260 12.3 THE STIFFNESS SYSTEM 260 12.4 PROPERTIES OF THE
STIFFNESS MATRIX 261 12.5 CALCULATING THE STIFFNESS MATRIX 263 12.6
EXAMPLE: RECTANGULAR DOMAIN AND UNIFORM MESH 266 12.7 M-MATRIX PROPERTY
IN THE ISOTROPIC CASE 269 12.8 HIGHLY ANISOTROPIC EQUATIONS 271 12.9
EXAMPLE: CIRCULAR DOMAIN 274 12.10 EXERCISES 276 12.11 ADVANCED
EXERCISES 277 13 UNSTRUCTURED FINITE-ELEMENT MESHES 281 13.1 CONCRETE
AND ABSTRACT OBJECTS 281 13.2 THE NODE OBJECT 284 13.3 THE
FINITE-ELEMENT OBJECT 286 13.4 THE MESH OBJECT 291 13.5 ASSEMBLING THE
STIFFNESS MATRIX 293 13.6 EXERCISES 296 14 ADAPTIVE MESH REFINEMENT 297
14.1 LOCAL REFINEMENT 297 14.2 ADAPTIVE REFINEMENT 299 14.3 THE
ADAPTIVE-REFINEMENT ALGORITHM 299 14.4 PRESERVING CONFORMITY 301 14.5
PRESERVING CONFORMITY IN PRACTICE 302 14.6 MESH REFINEMENT IN PRACTICE
304 14.7 AUTOMATIC BOUNDARY REFINEMENT 307 14.8 IMPLEMENTATION OF
AUTOMATIC BOUNDARY REFINEMENT 307 X CONTENTS 14.9 NONCONVEX DOMAINS 309
14.10 EXERCISES 312 15 HIGH-ORDER FINITE ELEMENTS 313 15.1 HIGH-ORDER
VS. LINEAR FINITE ELEMENTS 313 15.2 QUADRATIC FINITE ELEMENTS 314 15.3
CUBIC FINITE ELEMENTS 316 15.4 PROS AND CONS 317 15.5 EXERCISES 318 V
THE NUMERICAL SOLUTION OF LARGE SPARSE LINEAR SYSTEMS OF EQUATIONS 321
16 SPARSE MATRICES AND THEIR IMPLEMENTATION 325 16.1 SPARSE VS. DENSE
MATRICES 325 16.2 THE MATRIX-ELEMENT OBJECT 326 16.3 THE ROW OBJECT 329
16.4 THE SPARSE-MATRIX OBJECT 334 16.5 ASSEMBLING THE SPARSE STIFFNESS
MATRIX 337 16.6 EXERCISES 338 17 ITERATIVE METHODS FOR LARGE SPARSE
LINEAR SYSTEMS 341 17.1 ITERATIVE VS. DIRECT METHODS 341 17.2 ITERATIVE
METHODS 342 17.3 GAUSS-SEIDEL RELAXATION 344 17.4 JACOBI RELAXATION 344
17.5 SYMMETRIC GAUSS-SEIDEL 346 17.6 THE NORMAL EQUATION 346 17.7
INCOMPLETE FACTORIZATION 347 17.8 THE MULTIGRID METHOD 351 17.9
ALGEBRAIC MULTIGRID (AMG) 354 17.10 IMPLEMENTATION OF MULTIGRID 355
17.11 PRECONDITIONED CONJUGATE GRADIENTS (PCGS) 359 17.12 MULTIGRID FOR
NONSYMMETRIC LINEAR SYSTEMS 362 17.13 DOMAIN DECOMPOSITION AND MULTIGRID
362 17.14 EXERCISES 366 18 PARALLELISM 369 18.1 PARALLEL VS. SEQUENTIAL
COMPUTERS 369 18.2 PARALLELIZABLE ALGORITHMS 370 18.3 INHERENTLY
SEQUENTIAL ALGORITHMS 370 18.4 ARCHITECTURE 370 18.5 CACHE-ORIENTED
RELAXATION 371 18.6 SCHWARZ BLOCK RELAXATION 373 18.7 PARALLEL
ARCHITECTURES 373 18.8 SHARED MEMORY 373 CONTENTS XI 18.9 DISTRIBUTED
MEMORY 374 18.10 COMMUNICATION NETWORK 375 18.11 HYPERCUBE 376 18.12
EXAMPLE: MULTIPLICATION OF SPARSE MATRICES 378 18.13 LOW-LEVEL
C++IMPLEMENTATION 379 18.14 COMPARING ARCHITECTURES 380 18.15 JACOBI
RELAXATION 382 18.16 CONVERGENCE ANALYSIS 383 18.17 EXAMPLES FROM
HARWELL-BOEING 385 18.18 BLOCK JACOBI RELAXATION 388 18.19 EXERCISES 389
VI APPLICATIONS 391 19 DIFFUSION EQUATIONS 395 19.1 THE BOUNDARY-VALUE
PROBLEM 395 19.2 THE FINITE-ELEMENT MESH 396 19.3 THE LINEAR-SYSTEM
SOLVER 398 19.4 IMPLICIT TIME MARCHING 399 19.5 EXERCISES 401 20 THE
LINEAR ELASTICITY EQUATIONS 403 20.1 SYSTEM OF PDES 403 20.2 THE STRONG
FORMULATION 403 20.3 THE WEAK FORMULATION 404 20.4 THE MINIMIZATION
PROBLEM 405 20.5 COERCIVITY OF THE QUADRATIC FORM 405 20.6 THE
FINITE-ELEMENT DISCRETIZATION . 406 20.7 THE STIFFNESS MATRIX 407 20.8
THE ADAPTIVE-REFINEMENT CRITERION 408 20.9 THE MODIFIED MULTIGRID
ALGORITHM 408 20.10 NUMERICAL EXAMPLES 409 20.11 EXERCISES 412 21 THE
STOKES EQUATIONS 413 21.1 THE NABLA OPERATOR 413 21.2 GENERAL LINEAR
ELASTICITY 416 21.3 REDUCTION TO THE LINEAR ELASTICITY EQUATIONS 417
21.4 THE STOKES EQUATIONS 418 21.5 CONTINUATION PROCESS 418 21.6
ADEQUACY CONSIDERATION 419 21.7 PRECONDITIONER FOR THE STOKES EQUATIONS
419 21.8 FIRST-ORDER SYSTEM LEAST SQUARES 422 21.9 THE NAVIER-STOKES
EQUATIONS 423 21.10 EXERCISES 424 XII CONTENTS 22 ELECTROMAGNETIC WAVES
425 22.1 THE WAVE EQUATION 425 22.2 THE HELMHOLTZ EQUATION 426 22.3
FINITE-DIFFERENCE DISCRETIZATION 427 22.4 ADEQUACY IN FINITE DIFFERENCES
427 22.5 BILINEAR FINITE ELEMENTS 428 22.6 ADEQUACY IN BILINEAR FINITE
ELEMENTS 429 22.7 THE MEASURING PROBLEM 430 22.8 THE NEARLY RECTANGULAR
MODEL 431 22.9 THE NEARLY CIRCULAR MODEL 432 22.10 NONLINEAR FINITE
ELEMENTS 433 22.11 THE MAXWELL EQUATIONS 434 22.12 REDUCTION TO
THREE-DIMENSIONAL HELMHOLTZ EQUATIONS 436 22.13 EXERCISES 437 APPENDIX
439 A. 1 OPERATION S WITH VECTORS 439 A.2 OPERATIONS WITH MATRICES 440
A.3 OPERATIONS WITH DYNAMIC VECTORS 442 A.4 TWO-DIMENSIONAL
CONVECTION-DIFFUSION EQUATION 444 A.5 STABILITY IN THE EXPLICIT SCHEME
456 A.6 JORDAN FORM OF A TRIDIAGONAL MATRIX 456 A.7 DENOISING DIGITAL
IMAGES 458 A.8 MEMBERS OF THE MESH CLASS 460 A.9 OPERATIONS WITH SPARSE
MATRICES 462 A.10 KACMARZ ITERATION 465 A.LL ILU ITERATION 465 A. 12
MULTIGRID ITERATION 466 A. 13 ACCELERATION TECHNIQUES 470 A.14 PARALLEL
IMPLEMENTATION 475 A. 15 THE ADAPTIVE-REFINEMENT CODE 476 A.16 THE
DIFFUSION SOLVER 478 A.17 THE LINEAR ELASTICITY SOLVER 481 BIBLIOGRAPHY
485 INDEX 489 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T14:30:28Z |
| indexdate | 2024-07-09T20:38:21Z |
| institution | BVB |
| isbn | 0898716012 9780898716016 |
| language | English |
| lccn | 005054086 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014764033 |
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| physical | XXIII, 500 S. graph. Darst. |
| publishDate | 2006 |
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| spelling | Shapira, Yair Verfasser aut Solving PDEs in C++ numerical methods in a unified object-oriented approach Yair Shapira Philadelphia Society for Industrial and Applied Mathematics 2006 XXIII, 500 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Computational science and engineering Includes bibliographical references and index C++ (linguagem de programação) larpcal Equações diferenciais parciais larpcal Programação orientada a objetos larpcal C++ (Computer program language) Differential equations, Partial Object-oriented programming (Computer science) Objektorientierte Programmierung (DE-588)4233947-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf C++ (DE-588)4193909-8 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s Objektorientierte Programmierung (DE-588)4233947-9 s C++ (DE-588)4193909-8 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014764033&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shapira, Yair Solving PDEs in C++ numerical methods in a unified object-oriented approach C++ (linguagem de programação) larpcal Equações diferenciais parciais larpcal Programação orientada a objetos larpcal C++ (Computer program language) Differential equations, Partial Object-oriented programming (Computer science) Objektorientierte Programmierung (DE-588)4233947-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd C++ (DE-588)4193909-8 gnd |
| subject_GND | (DE-588)4233947-9 (DE-588)4128130-5 (DE-588)4044779-0 (DE-588)4193909-8 |
| title | Solving PDEs in C++ numerical methods in a unified object-oriented approach |
| title_auth | Solving PDEs in C++ numerical methods in a unified object-oriented approach |
| title_exact_search | Solving PDEs in C++ numerical methods in a unified object-oriented approach |
| title_exact_search_txtP | Solving PDEs in C++ numerical methods in a unified object-oriented approach |
| title_full | Solving PDEs in C++ numerical methods in a unified object-oriented approach Yair Shapira |
| title_fullStr | Solving PDEs in C++ numerical methods in a unified object-oriented approach Yair Shapira |
| title_full_unstemmed | Solving PDEs in C++ numerical methods in a unified object-oriented approach Yair Shapira |
| title_short | Solving PDEs in C++ |
| title_sort | solving pdes in c numerical methods in a unified object oriented approach |
| title_sub | numerical methods in a unified object-oriented approach |
| topic | C++ (linguagem de programação) larpcal Equações diferenciais parciais larpcal Programação orientada a objetos larpcal C++ (Computer program language) Differential equations, Partial Object-oriented programming (Computer science) Objektorientierte Programmierung (DE-588)4233947-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd C++ (DE-588)4193909-8 gnd |
| topic_facet | C++ (linguagem de programação) Equações diferenciais parciais Programação orientada a objetos C++ (Computer program language) Differential equations, Partial Object-oriented programming (Computer science) Objektorientierte Programmierung Numerisches Verfahren Partielle Differentialgleichung C++ |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014764033&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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