Construction Of Integration Formulas For Initial Value Problems

1st Edition - January 1, 1977
Author: P.J. Van Der Houwen
Language: English
eBook ISBN:
9 7 8 - 0 - 4 4 4 - 6 0 1 8 9 - 6

Construction of Integration Formulas for Initial Value Problems provides practice-oriented insights into the numerical integration of initial value problems for ordinary… Read more

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Construction of Integration Formulas for Initial Value Problems provides practice-oriented insights into the numerical integration of initial value problems for ordinary differential equations. It describes a number of integration techniques, including single-step methods such as Taylor methods, Runge-Kutta methods, and generalized Runge-Kutta methods. It also looks at multistep methods and stability polynomials. Comprised of four chapters, this volume begins with an overview of definitions of important concepts and theorems that are relevant to the construction of numerical integration methods for initial value problems. It then turns to a discussion of how to convert two-point and initial boundary value problems for partial differential equations into initial value problems for ordinary differential equations. The reader is also introduced to stiff differential equations, partial differential equations, matrix theory and functional analysis, and non-linear equations. The order of approximation of the single-step methods to the differential equation is considered, along with the convergence of a consistent single-step method. There is an explanation on how to construct integration formulas with adaptive stability functions and how to derive the most important stability polynomials. Finally, the book examines the consistency, convergence, and stability conditions for multistep methods. This book is a valuable resource for anyone who is acquainted with introductory calculus, linear algebra, and functional analysis.

PrefaceChapter 1. Introduction 1.1. Initial Value Problems 1.1.1. Definitions 1.1.2. Reduction to autonomous, first order form 1.1.3. Existence and uniqueness 1.2. Two-Point Boundary Value Problems 1.3. Stiff Differential Equations 1.3.1. Geometrical definition of a stiff differential equation 1.3.2. The Jacobian matrix of a stiff differential equation 1.4. Partial Differential Equations 1.4.1. Initial-boundary value problems 1.4.2. The method of lines 1.5. Some Matrix Theory and Functional Analysis 1.5.1. Notations and definitions 1.5.2. Theorems 1.6. Difference Schemes 1.6.1. Preliminary definitions 1.6.2. Consistency 1.6.3. Convergence 1.6.4. Stability 1.6.5. Interpolation formulas 1.7. Non-Linear Equations 1.7.1. Iteration functions 1.7.2. Jacobi iteration 1.7.3. Accelerated Jacobi iteration 1.7.4. Richardson iteration 1.7.5. Newton-Raphson iterationChapter 2. Single-Step Methods 2.1. Taylor Methods 2.1.1 General considerations 2.1.2. Implicit Taylor methods 2.2. Runge-Kutta Methods 2.3. Generalized Runge-Kutta Methods 2.3.1. Formulas based on one evaluation of the Jacobian matrix 2.3.2. Formulas based on several evaluations of the Jacobian matrix 2.4. Consistency Conditions 2.4.1. Taylor methods 2.4.2. Iterated Taylor methods 2.4.3. Taylor-Runge-Kutta formulas 2.4.4. Formulas of Liniger and Willoughby 2.4.5. Runge-Kutta methods 2.4.6. Generalized Runge-Kutta methods 2.5. Convergence Conditions 2.6. Stability Conditions 2.6.1. Propagation of numerical errors 2.6.2. The stability function 2.6.3. The stability region 2.6.4. Stability functions for special problems 2.6.5. Pade approximations 2.6.6. Polynomials with optimal real stability boundary 2.6.7. Polynomials with optimal imaginary stability boundary 2.6.8. Polynomials with widely separated subregions of stability 2.6.9. Exponentially fitted functions 2.6.10. The local numerical error in case of Runge-Kutta methods 2.7. Construction of Integration Formulas with Adaptive Stability Functions 2.7.1. Taylor formulas 2.7.2. First order Runge-Kutta formulas 2.7.3. Second order Runge-Kutta formulas 2.7.4. Third order Runge-Kutta formulas 2.7.5. Fourth order Runge-Kutta formulas 2.7.6. Generalized Runge-Kutta formulas of first and second order 2.7.7. Generalized Runge-Kutta formulas of third order 2.7.8. Generalized Runge-Kutta formulas of fourth order 2.8. Numerical Experiments 2.8.1. Methods used 2.8.2. Computational results 2.8.3. Conclusions 2.9. Step Size Control 2.9.1. The discrepance of the difference solution 2.9.2. Error estimates for Taylor methods 2.9.3. Error estimates for Runge-Kutta methods 2.9.4. The residual term 2.9.5. The discrepance of linearity 2.9.6. Numerical examples 2.9.7. ConclusionsChapter 3. Multistep Methods 3.1. Multistep Taylor Methods 3.1.1. Linear multistep methods 3.1.2. Implicit linear multistep methods 3.1.3. Higher derivative forms 3.2. Multistep Runge-Kutta Methods 3.3. Generalized Multistep Runge-Kutta Methods 3.4. Consistency Conditions 3.4.1. Linear multistep methods 3.4.2. Two-point multistep Runge-Kutta methods 3.4.3. Multipoint two-step Runge-Kutta methods 3.4.4. Generalized linear multistep methods 3.5. Convergence Conditions 3.5.1. A necessary condition 3.5.2. A sufficient condition 3.6. Stability Conditions 3.6.1. The characteristic equation 3.6.2. The stability region 3.7. Two-Step Runge-Kutta Methods with Extended Real Stability Interval 3.7.1. The stability condition 3.7.2. First order formulas using m stages 3.7.3. Second order formulas using m stages 3.7.4. Third order formulas using three stages 3.7.5. Fourth order formulas using four stages 3.8. Generalized Linear Multistep Methods with Adaptive Principal Root and Zero-Parasitic Roots 3.8.1. Two-step formulas 3.8.2. Two-step formulas with zero-parasitic root 3.8.3. Higher-step formulas with zero-parasitic rootsChapter 4. Stability Polynomials 4.1. The Polynomials Pm, 0 4.2. The Polynomials Rm(p) 4.2.1. Definition 4.2.2. The 'equal ripple' property 4.2.3. Approximation by Chebyshev polynomials 4.2.4. Approximation by least squares 4.2.5. Numerical approximation of the polynomials Rm(p) 4.2.6. Strongly stable modifications of the polynomials Rm(p) 4.2.7. Polynomials of changing order and degree 4.3. The Polynomials Im(p) 4.3.1. Definition 4.3.2. Some direct results 4.3.3. An upper bound for ß 4.3.4. The optimal polynomials for odd values of m 4.4. The Polynomials Zm(p) 4.4.1. Statement of the problem 4.4.2. Derivation of the optimal polynomials 4.4.3. Stability regionsBibliographyIndex