Fatunla / Rheinboldt / Siewiorek Numerical Methods for Initial Value Problems in Ordinary Differential Equations

1. Auflage 2014
ISBN: 978-1-4832-6926-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 308 Seiten, Web PDF

Numerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular, and discontinuous initial value problems, characterized by large Lipschitz constants. The book reviews the difference operators, the theory of interpolation, first integral mean value theorem, and numerical integration algorithms. The text explains the theory of one-step methods, the Euler scheme, the inverse Euler scheme, and also Richardson's extrapolation. The book discusses the general theory of Runge-Kutta processes, including the error estimation, and stepsize selection of the R-K process. The text evaluates the different linear multistep methods such as the explicit linear multistep methods (Adams-Bashforth, 1883), the implicit linear multistep methods (Adams-Moulton scheme, 1926), and the general theory of linear multistep methods. The book also reviews the existing stiff codes based on the implicit/semi-implicit, singly/diagonally implicit Runge-Kutta schemes, the backward differentiation formulas, the second derivative formulas, as well as the related extrapolation processes. The text is intended for undergraduates in mathematics, computer science, or engineering courses, andfor postgraduate students or researchers in related disciplines.

Autoren/Hrsg.

Fatunla, Simeon Ola

Rheinboldt, Werner

Siewiorek, Daniel

Inhaltsverzeichnis

1;Front Cover;1
2;Numerical Methods for Initial Value Problems in Ordinary Differential Equations;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;CHAPTER 1. PRELIMINARIES;14
6.1;1.1 The Difference Operators;14
6.2;1.2 Theory of Interpolation;16
6.3;1.3 Finite Difference Equations;23
6.4;1.4 Linear Systems with Constant Coefficients;26
6.5;1.5 Distribution of Roots of Polynomials;28
6.6;1.6 First Integral Mean Value Theorem;30
6.7;1.7 Common Norms in ODEs;31
7;CHAPTER 2. NUMERICAL INTEGRATION ALGORITHMS;33
7.1;2.1 Introduction;33
7.2;2.2 Existence of Solution, Numerical Approach;36
7.3;2.3 Special IVPs;39
7.4;2.4 Error Propagation, Stability and Convergence of Discretization Methods;41
8;CHAPTER 3. THEORY OF ONE-STEP METHODS;44
8.1;3.1 General Theory of One-Step Methods;44
8.2;3.2 The Euler Scheme, the Inverse Euler Scheme and Richardson's Extrapolation;48
8.3;3.3 The Convergence of Euler's Scheme;52
8.4;3.4 The Trapezoidal Scheme;55
9;CHAPTER 4. RUNGE-KUTTA PROCESSES;61
9.1;4.1 General Theory of Runge-Kutta Processes;61
9.2;4.2 The Explicit Two-Stage Process;69
9.3;4.3 Convergence and Stability of Two-Stage Explicit R-K Scheme;73
9.4;4.4 Matrix Representation of the R-K Processes;75
9.5;4.5 Error Estimation and Stepsize Selection in R-K Processes;85
9.6;4.6 Implicit and Semi-Implicit R-K Processes;88
9.7;4.7 Rosenbrock Methods;98
10;CHAPTER 5. LINEAR MULTISTEP METHODS;102
10.1;5.1 Starting Procedure;102
10.2;5.2 Explicit Linear Multistep Methods (Adams-Bashforth 1883,
and related methods);104
10.3;5.3 Implicit Linear Multistep Methods (Adams-Moulton scheme, 1926 and
related methods);108
10.4;5.4 Implementation of the Predictor-Corrector Formulas;113
10.5;5.5 General Theory of Linear Multistep Methods;117
10.6;5.6 Automatic Implementation of the Adams Scheme;134
11;CHAPTER 6. NUMERICAL TREATMENT OF SINGULAR/ DISCONTINUOUS INITIAL VALUE PROBLEMS;138
11.1;6.1 Introduction;138
11.2;6.2 Non-Polynomial Methods;140
11.3;6.3 The Inverse Polynomial Methods;146
11.4;6.4 Local Error Estimates in Automatic Codes for Discontinuous Systems;149
12;CHAPTER 7. EXTRAPOLATION PROCESSES AND SINGULARITIES;153
12.1;7.1 Introduction;153
12.2;7.2 Generation of the Zero-th Column of Extrapolation Table;155
12.3;7.3 Polynomial and Rational Extrapolation;161
12.4;7.4 Convergence and Stability Properties of Extrapolation Processes;167
12.5;7.5 Practical Implementation of Extrapolation Processes;171
13;CHAPTER 8. STIFF INITIAL VALUE PROBLEMS;174
13.1;8.1 The Concept of Stiffness;174
13.2;8.2 Stiff and Nonstiff Algorithms;180
13.3;8.3 Solution of Nonlinear Equations and Estimation of Jacobians;181
13.4;8.4 Region of Absolute Stability;185
13.5;8.5 Stability Criteria for Stiff Methods;193
13.6;8.6 Stronger Stability Properties of IRK Processes;201
13.7;8.7 One-Leg Multistep Methods;208
14;CHAPTER 9. STIFF ALGORITHMS;211
14.1;9.1 What are Stiff Algorithms?;211
14.2;9.2 Efficient Implementation of Implicit Runge-Kutta Methods (IRK);216
14.3;9.3 The Backward Differentiation Formula (BDF);224
14.4;9.4 Second Derivative Formulas (SDFs);228
14.5;9.5 Extrapolation Processes for Stiff Systems;230
14.6;9.6 Mono-Implicit R-K Formulas;236
15;CHAPTER 10. SECOND ORDER DIFFERENTIAL EQUATIONS;239
15.1;10.1 Introduction;239
15.2;10.2 Linear Multistep Methods and the Concept of P-Stability;241
15.3;10.3 Derivation of P-Stable Formulas;243
15.4;10.4 One-Leg Multistep Methods for Second Order IVPs;245
15.5;10.5 Multiderivative Methods for Second IVPs;249
16;CHAPTER 11. RECENT DEVELOPMENTS IN ODE SOLVERS;251
17;REFERENCES ;266
18;INDEX;300