Numerical solution of time dependent advection diffusion reaction equations:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2003
|
| Series: | Springer series in computational mathematics
33 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [447] - 464 |
| Physical Description: | X, 471 S. graph. Darst. |
| ISBN: | 3540034404 |
Staff View
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| 100 | 1 | |a Hundsdorfer, Willem Hans |d 1954- |e Verfasser |0 (DE-588)141220120 |4 aut | |
| 245 | 1 | 0 | |a Numerical solution of time dependent advection diffusion reaction equations |c Willem Hundsdorfer ; Jan Verwer |
| 246 | 1 | 3 | |a Numerical solution of time-dependent advection-diffusion-reaction equations |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a X, 471 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer series in computational mathematics |v 33 | |
| 500 | |a Literaturverz. S. [447] - 464 | ||
| 650 | 4 | |a Advektion-Diffusionsgleichung - Reaktions-Diffusionsgleichung - Numerisches Verfahren | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
| 650 | 4 | |a Runge-Kutta formulas | |
| 650 | 4 | |a Stiff computation (Differential equations) | |
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Record in the Search Index
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| adam_text | Table of Contents
I Basic Concepts and Discretizations 1
1 Advection Diffusion Reaction Equations 1
1.1 Nonlinear Reaction Problems from Chemistry 3
1.2 Model Equations for Advection Diffusion 9
1.3 Multi dimensional Problems 14
1.4 Examples of Applications 18
2 Basic Discretizations for ODEs 23
2.1 Initial Value Problems and Euler s Method 23
2.2 Norms and Matrices 27
2.3 Perturbations on ODE Systems 30
2.4 The 0 Method and Stiff Problems 35
2.5 Stability of the 0 Method 37
2.6 Consistency and Convergence of the 0 Method 42
2.7 Nonlinear Results for the 0 Method 44
2.8 Concluding Remarks 46
3 Basic Spatial Discretizations 48
3.1 Discrete Fourier Decompositions 49
3.2 The Advection Equation 52
3.3 The Diffusion Equation 62
3.4 The Advection Diffusion Equation 66
4 Convergence of Spatial Discretizations 71
4.1 Stability, Consistency and Convergence 71
4.2 Advection Diffusion with Constant Coefficients 74
4.3 Advection with Variable Coefficients 77
4.4 Diffusion with Variable Coefficients 81
4.5 Variable Coefficients and Higher Order Schemes 83
5 Boundary Conditions and Spatial Accuracy 84
5.1 Refined Global Error Estimates 85
5.2 Outflow with Central Advection Discretization 86
5.3 Boundary Conditions with the Heat Equation 88
5.4 Boundary Conditions and Higher Order Schemes 92
6 Time Stepping for PDEs 94
6.1 The Method of Lines and Direct Discretizations 94
6.2 Stability, Consistency and Convergence 99
VIII Table of Contents
6.3 Stability for MOL Stability Regions 103
6.4 Von Neumann Stability Analysis Ill
7 Monotonicity Properties 116
7.1 Positivity and Maximum Principle 116
7.2 Positive Semi discrete Systems 118
7.3 Positive Time Stepping Methods 121
7.4 Numerical Illustrations 124
8 Numerical Test Examples 127
8.1 The Nonlinear Schrodinger Equation 128
8.2 The Angiogenesis Model 134
II Time Integration Methods 139
1 Runge Kutta Methods 139
1.1 The Order Conditions 140
1.2 Examples 142
1.3 The Stability Function 144
1.4 Step Size Restrictions for Advection Diffusion 149
1.5 Rosenbrock Methods 151
2 Convergence of Runge Kutta Methods 155
2.1 Order Reduction 155
2.2 Local Error Analysis 158
2.3 Global Error Analysis 161
2.4 Concluding Notes 166
3 Linear Multistep Methods 170
3.1 The Order Conditions 171
3.2 Examples 173
3.3 Stability Analysis 174
3.4 Step Size Restrictions for Advection Diffusion 181
3.5 Convergence Analysis 182
4 Monotone ODE Methods 185
4.1 Linear Positivity for One Step Methods 185
4.2 Nonlinear Positivity for One Step Methods 189
4.3 Positivity for Multistep Methods 192
4.4 Related Monotonicity Results 196
5 Variable Step Size Control 197
5.1 Step Size Selection 197
5.2 An Explicit Runge Kutta Example 200
5.3 An Implicit Multistep Example 203
5.4 General Purpose ODE Codes 205
6 Numerical Examples 206
6.1 A Model for Antibodies in Tumorous Tissue 206
6.2 The Nonlinear Schrodinger Equation 209
Table of Contents IX
III Advection Diffusion Discretizations 215
1 Non oscillatory MOL Advection Discretizations 215
1.1 Spatial Discretization for Linear Advection 215
1.2 Numerical Examples 222
1.3 Positivity and the TVD Property 226
1.4 Nonlinear Scalar Conservation Laws 233
2 Direct Space Time Advection Discretizations 239
2.1 Optimal Order DST Schemes 239
2.2 A Non oscillatory Third Order DST Scheme 243
2.3 Explicit Schemes with Unconditional Stability 248
3 Implicit Spatial Discretizations 250
3.1 Order Conditions 251
3.2 Examples 253
3.3 Stability and Convergence 258
3.4 Monotonicity 261
3.5 Time Integration Aspects 263
4 Non uniform Grids Finite Volumes (ID) 264
4.1 Vertex Centered Schemes 265
4.2 Cell Centered Schemes 272
4.3 Numerical Illustrations 278
4.4 Higher Order Methods and Limiting 281
5 Non uniform Grids Finite Elements (ID) 283
5.1 The Basic Galerkin Method 283
5.2 Standard Galerkin Error Estimates 288
5.3 Upwinding 291
6 Multi dimensional Aspects 292
6.1 Cartesian Grid Discretizations 293
6.2 Diffusion on Cartesian Grids 295
6.3 Advection on Cartesian Grids 303
6.4 Transformed Cartesian Grids 308
6.5 Unstructured Grids 311
7 Notes on Moving Grids and Grid Refinement 316
7.1 Dynamic Regridding 316
7.2 Static Regridding 321
IV Splitting Methods 325
1 Operator Splitting 325
1.1 First Order Splitting 325
1.2 Second Order Symmetrical Splitting 329
1.3 Higher Order Splittings 330
1.4 Abstract Initial Value Problems 331
1.5 Advection Diffusion Reaction Splittings 335
1.6 Dimension Splitting 337
1.7 Boundary Values and Stiff Terms 344
2 LOD Methods 348
X Table of Contents
2.1 The LOD Backward Euler Method 348
2.2 LOD Crank Nicolson Methods 351
2.3 The Trapezoidal Splitting Method 359
2.4 Boundary Correction Techniques 365
2.5 Numerical Comparisons 367
3 ADI Methods 369
3.1 The Peaceman Rachford Method 369
3.2 The Douglas Method 373
4 IMEX Methods 383
4.1 The IMEX 0 Method 383
4.2 IMEX Multistep Methods 386
4.3 Notes on IMEX Runge Kutta Methods 391
4.4 Concluding Remarks and Tests 393
5 Rosenbrock AMF Methods 398
5.1 One Stage Methods of Order One and Two 398
5.2 Two Stage Methods of Order Two and Three 400
5.3 A Three Stage Method of Order Two 403
5.4 Concluding Remarks and Tests 405
6 Numerical Examples 409
6.1 Two Chemo taxis Problems from Biology 409
6.2 The Numerical Methods 411
6.3 Numerical Experiments 412
V Stabilized Explicit Runge Kutta Methods 419
1 The RKC Family 420
1.1 Stability Polynomials 420
1.2 Integration Formulas 426
1.3 Internal Stability and Full Convergence Properties .... 430
2 The ROCK Family 433
2.1 Stability Polynomials 433
2.2 Integration Formulas 435
2.3 Internal Stability and Convergence 436
3 Numerical Examples 438
3.1 A Combustion Model 439
3.2 A Radiation Diffusion Model 441
Bibliography 447
Index 465
|
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| dewey-ones | 515 - Analysis |
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| spelling | Hundsdorfer, Willem Hans 1954- Verfasser (DE-588)141220120 aut Numerical solution of time dependent advection diffusion reaction equations Willem Hundsdorfer ; Jan Verwer Numerical solution of time-dependent advection-diffusion-reaction equations Berlin [u.a.] Springer 2003 X, 471 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 33 Literaturverz. S. [447] - 464 Advektion-Diffusionsgleichung - Reaktions-Diffusionsgleichung - Numerisches Verfahren Differential equations Numerical solutions Differential equations, Partial Numerical solutions Runge-Kutta formulas Stiff computation (Differential equations) Advektion-Diffusionsgleichung (DE-588)4580212-9 gnd rswk-swf Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Advektion-Diffusionsgleichung (DE-588)4580212-9 s Reaktions-Diffusionsgleichung (DE-588)4323967-5 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Verwer, Jan Verfasser aut Springer series in computational mathematics 33 (DE-604)BV000012004 33 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010441274&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hundsdorfer, Willem Hans 1954- Verwer, Jan Numerical solution of time dependent advection diffusion reaction equations Springer series in computational mathematics Advektion-Diffusionsgleichung - Reaktions-Diffusionsgleichung - Numerisches Verfahren Differential equations Numerical solutions Differential equations, Partial Numerical solutions Runge-Kutta formulas Stiff computation (Differential equations) Advektion-Diffusionsgleichung (DE-588)4580212-9 gnd Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4580212-9 (DE-588)4323967-5 (DE-588)4128130-5 |
| title | Numerical solution of time dependent advection diffusion reaction equations |
| title_alt | Numerical solution of time-dependent advection-diffusion-reaction equations |
| title_auth | Numerical solution of time dependent advection diffusion reaction equations |
| title_exact_search | Numerical solution of time dependent advection diffusion reaction equations |
| title_full | Numerical solution of time dependent advection diffusion reaction equations Willem Hundsdorfer ; Jan Verwer |
| title_fullStr | Numerical solution of time dependent advection diffusion reaction equations Willem Hundsdorfer ; Jan Verwer |
| title_full_unstemmed | Numerical solution of time dependent advection diffusion reaction equations Willem Hundsdorfer ; Jan Verwer |
| title_short | Numerical solution of time dependent advection diffusion reaction equations |
| title_sort | numerical solution of time dependent advection diffusion reaction equations |
| topic | Advektion-Diffusionsgleichung - Reaktions-Diffusionsgleichung - Numerisches Verfahren Differential equations Numerical solutions Differential equations, Partial Numerical solutions Runge-Kutta formulas Stiff computation (Differential equations) Advektion-Diffusionsgleichung (DE-588)4580212-9 gnd Reaktions-Diffusionsgleichung (DE-588)4323967-5 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Advektion-Diffusionsgleichung - Reaktions-Diffusionsgleichung - Numerisches Verfahren Differential equations Numerical solutions Differential equations, Partial Numerical solutions Runge-Kutta formulas Stiff computation (Differential equations) Advektion-Diffusionsgleichung Reaktions-Diffusionsgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010441274&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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