Iterative Splitting Methods for Differential Equations (Chapman and Hall CRC Numerical Analysis and Scientific Computation Series)

Contents......Page 10
Preface......Page 8
Introduction......Page 25
1. Model Problems......Page 31
1.1 Related Models for Decomposition......Page 32
1.2.1 Waste Disposal......Page 33
1.2.2 Elastic Wave Propagation......Page 35
1.2.3 Deposition Models: CVD (Chemical Vapor Deposition) Processes......Page 37
1.2.4 Navier-Stokes Molecular Dynamics: Coupling Continuous and Discrete Problems......Page 44
2.1 Historical Overview......Page 49
2.2 Decomposition Ideas......Page 50
2.2.1 Physical Decomposition......Page 51
2.2.2 Mathematical Decomposition......Page 53
2.3 Introduction to Classical Splitting Methods......Page 54
2.3.2 Sequential Splitting Method......Page 55
2.3.3 Symmetrical Weighted Sequential Splitting Methods......Page 57
2.3.4 Strang-Marchuk Splitting Method......Page 58
2.3.5 Higher-Order Splitting Methods......Page 59
2.4 Iterative Splitting Method......Page 61
2.5.1 Local Error Analysis......Page 64
2.5.2 Increasing the Order of Accuracy with Improved Initial Functions and Consistency Analysis......Page 68
2.6 Stability Analysis of the Iterative Splitting Method for Bounded Operators......Page 71
2.6.1 Time Integration Methods......Page 72
2.6.2 Time Discretization Methods......Page 76
3. Decomposition Methods for Partial Differential Equations......Page 83
3.1 Iterative Schemes for Unbounded Operators......Page 84
3.1.1 Iterative Splitting Schemes......Page 85
3.1.2 One-Stage Iterative Splitting Method for A-bounded Operators......Page 86
3.1.3 Two-Stage Iterative Schemes for Operators Generating an Analytical Semigroup......Page 91
3.1.4 Some Examples for One-Stage and Two-Stage Iterative Operator Splitting Schemes......Page 97
4.1 Exponential Runge-Kutta Methods to Compute Iterative Splitting Schemes......Page 101
4.2 Matrix Exponentials to Compute Iterative Splitting Schemes......Page 103
4.2.1 Derivation of the Formula......Page 104
4.3 Algorithms......Page 105
4.3.2 One-Side Scheme (Alternative Notation with Commutators)......Page 106
5. Extensions of Iterative Splitting Schemes......Page 109
5.1.1 Balancing of Time and Spatial Discretization......Page 110
5.1.2 Spatial Discretization Schemes with Dimensional Splitting......Page 111
5.2.1 Combined Time-Space Iterative Splitting Method......Page 117
5.2.2 Nonoverlapping Time-Space Iterative Splitting Method......Page 119
5.2.3 Overlapping Time-Space Iterative Splitting Method......Page 120
5.2.4 Error Analysis and Convergence of Combined Method......Page 121
5.3 Successive Approximation for Time-Dependent Operators......Page 125
5.3.1 Algorithm for Successive Approximation......Page 126
6.1 Introduction......Page 129
6.2.1 Introduction Problem 1: Starting Conditions......Page 132
6.2.2 Introduction Problem 2: Stiffness of Matrices......Page 136
6.2.3 Introduction Problem 3: Nonsplitting and Splitting......Page 139
6.2.4 Introduction Problem 4: System of ODEs with Stiff and Nonstiff Cases......Page 141
6.2.5 Introduction Problem 5: Linear Partial Differential Equation......Page 146
6.2.6 Introduction Problem 6: Nonlinear Ordinary Differential Equation......Page 148
6.2.7 Introduction Problem 7: Coupling Convection-Diffusion and Reaction Equations with Separate Codes......Page 150
6.3.1 Comparison Problem 1: Iterative Splitting Method with Improved Time Discretization Methods......Page 154
6.3.2 Comparison Problem 2: Iterative Splitting Method Compared to Standard Splitting Methods......Page 162
6.4.1 Extension Problem1: Spatial DecompositionMethods (Classical and Iterative Splitting Schemes)......Page 173
6.4.2 Extension Problem 2: Hyperbolic Equations......Page 183
6.4.3 Extension Problem3: Nonlinear Partial Differential Equations......Page 198
6.4.6 Second Experiment: 10 ×10Matrix......Page 210
6.4.7 Third Example: Commutator Problem......Page 211
6.4.8 Two-Phase Example......Page 213
6.5.1 Waste Disposal: Transport and Reaction of Radioactive Contaminants......Page 226
6.5.2 Elastic Wave Propagation......Page 233
6.5.3 CVD Apparatus: Optimization of a Deposition Problem......Page 245
6.5.4 Complex Flow Phenomena: Navier-Stokes and Molecular Dynamics......Page 270
6.6 Conclusion to Numerical Experiments: Discussion of Some Delicate Problems......Page 280
7. Summary and Perspectives......Page 283
8.1.1 Rough Structuring of the Software Packages......Page 285
8.1.2 UG Software Toolbox......Page 286
8.1.3 UG Concept......Page 287
8.1.5 Equations in d3f......Page 289
8.1.6 Structure of d3f......Page 290
8.2.2 Task of r3t......Page 291
8.2.3 Conception of r3t......Page 292
8.2.4 Application of r3t......Page 293
8.3 Solving PDEs Using FIDOS......Page 294
8.3.1 PDEs Treated......Page 295
8.3.2 Methods......Page 297
8.3.3 Iterative Operator Splitting Methods......Page 298
8.3.4 EigenvalueMethods......Page 299
8.3.5 Numerical Examples......Page 300
List of Abbreviations......Page 305
Symbols......Page 306
General Notations......Page 308
Bibliography......Page 309
F......Page 325
P......Page 326
W......Page 327