Numerical Time-Dependent Partial Differential Equations for Scientists and Engineers (eBook)
312 Seiten
Elsevier Science (Verlag)
978-0-08-091704-7 (ISBN)
The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs, methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them.
In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions, Practical stability analysis in the presence of the boundaries and interfaces, Treatment of problems with different temporal/spatial scales either explicit or implicit, preservation of symmetries and additional constraints, physical regularization of singularities, resolution enhancement using adaptive mesh refinement and moving meshes.
- Self contained presentation of key issues in successful numerical simulation
- Accessible to scientists and engineers with diverse background
- Provides analysis of the dispersion relation, symmetries, particular solutions and instabilities of the partial differential equations
It is the first text that in addition to standard convergence theory treats other necessary ingredients for successful numerical simulations of physical systems encountered by every practitioner. The book is aimed at users with interests ranging from application modeling to numerical analysis and scientific software development. It is strongly influenced by the authors research in in space physics, electrical and optical engineering, applied mathematics, numerical analysis and professional software development. The material is based on a year-long graduate course taught at the University of Arizona since 1989. The book covers the first two-semesters of a three semester series. The second semester is based on a semester-long project, while the third semester requirement consists of a particular methods course in specific disciplines like computational fluid dynamics, finite element method in mechanical engineering, computational physics, biology, chemistry, photonics, etc.The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs; methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them.In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions; Practical stability analysis in the presence of the boundaries and interfaces; Treatment of problems with different temporal/spatial scales either explicit or implicit; preservation of symmetries and additional constraints; physical regularization of singularities; resolution enhancement using adaptive mesh refinement and moving meshes. - Self contained presentation of key issues in successful numerical simulation- Accessible to scientists and engineers with diverse background- Provides analysis of the dispersion relation, symmetries, particular solutions and instabilities of the partial differential equations
Front cover 1
Half title page 2
Copyright page 5
Preface 6
Contents 10
Chapter 1. Overview of Partial Differential Equations 12
1.1. Examples of Partial Differential Equations 12
1.2. Linearization and Dispersion Relation 18
1.3. Well-posedness, Regularity and the Solution Operator 26
1.4. Physical Instabilities 31
1.5. Group Velocity, Wave Action and Wave Energy Equations 56
1.6. Project Assignment 63
1.7. Project Sample 64
Chapter 2. Discretization Methods 70
2.1. Polynomial Interpolation and Finite Differences 71
2.2. Compact Finite Differences and Dispersion Preserving Schemes 85
2.3. Spectral Differentiation 90
2.4. Method of Weighted Residuals, Finite Element and Finite Volume Methods 103
2.5. Project Assignment 112
2.6. Project Sample 113
Chapter 3. Convergence Theory for Initial Value Problems 120
3.1. Introduction to Convergence Theory 120
3.2. Lax-Richtmyer Equivalence Theorem 128
3.3. Von Neumann Analysis and Courant-Friedrichs-Levy Necessary Stability Condition 141
3.4. Project Assignment 150
3.5. Project Sample 151
Chapter 4. Numerical Boundary Conditions 156
4.1. Introduction to Numerical Boundary and Interface Conditions 156
4.2. Transparent Boundary Conditions for Hyperbolic and Dispersive Systems 158
4.3. Berenger's Perfectly Matched Layer Boundary Conditions 166
4.4. Matrix Stability Analysis in the Presence of Boundaries and Interfaces 176
4.5. Project Sample 179
Chapter 5. Problems with Multiple Temporal and Spatial Scales 186
5.1. Examples of Weakly and Strongly Interacting Multiple Scales 186
5.2. Stiff Ordinary Differential Equation Solvers 198
5.3. Long-Time Integrators for Hamiltonian Systems 201
5.4. Hyperbolic Conservation Laws 221
5.5. Methods of Fractional Steps, Time-Split and Approximate Factorization Algorithms 251
5.6. Project Sample 256
Chapter 6. Numerical Grid Generation 262
6.1. Non-uniform Static Grids, Stability and Accuracy Issues 262
6.2. Adaptive and Moving Grids Based on Equidistribution Principle 267
6.3. Level Set Methods 269
6.4. The Front Tracking Method 274
6.5. Project Sample 279
Bibliography 284
Index 300
Recent titles 306
Erscheint lt. Verlag | 21.9.2010 |
---|---|
Sprache | englisch |
Themenwelt | Sachbuch/Ratgeber |
Mathematik / Informatik ► Informatik ► Theorie / Studium | |
Mathematik / Informatik ► Mathematik ► Analysis | |
Technik ► Bauwesen | |
ISBN-10 | 0-08-091704-6 / 0080917046 |
ISBN-13 | 978-0-08-091704-7 / 9780080917047 |
Haben Sie eine Frage zum Produkt? |
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