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Numerical Methods for Ordinary Differential Equations : Initial Value Problems / by David F. Griffiths, Desmond J. Higham
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Undergraduate Mathematics Series | SpringerLink BücherPublisher: London : Springer-Verlag London Limited, 2010Description: Online-Ressource (XIV, 271p. 69 illus, digital)ISBN:- 9780857291486
- 518
- QA297-299.4
- QA371.3
Contents:
Summary: ODEs—An Introduction -- Euler’s Method -- The Taylor Series Method -- Linear Multistep Methods—I: Construction and Consistency -- Linear Multistep Methods—II: Convergence and Zero-Stability -- Linear Multistep Methods—III: Absolute Stability -- Linear Multistep Methods—IV: Systems of ODEs -- Linear Multistep Methods—V: Solving Implicit Methods -- Runge–Kutta Method—I: Order Conditions -- Runge-Kutta Methods–II Absolute Stability -- Adaptive Step Size Selection -- Long-Term Dynamics -- Modified Equations -- Geometric Integration Part I—Invariants -- Geometric Integration Part II—Hamiltonian Dynamics -- Stochastic Differential Equations.Summary: Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors. The book covers key foundation topics: o Taylor series methods o Runge-Kutta methods o Linear multistep methods o Convergence o Stability and a range of modern themes: o Adaptive stepsize selection o Long term dynamics o Modified equations o Geometric integration o Stochastic differential equations The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com.PPN: PPN: 1650602049Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Contents""; ""1 ODEs�An Introduction""; ""1.1 Systems of ODEs""; ""1.2 Higher Order Differential Equations""; ""1.3 Some Model Problems""; ""2 Euler�s Method""; ""2.1 A Preliminary Example""; ""2.1.1 Analysing the Numbers""; ""2.2 Landau Notation""; ""2.3 The General Case""; ""2.4 Analysing the Method""; ""2.5 Application to Systems""; ""3 The Taylor Series Method""; ""3.1 Introduction""; ""3.2 An Order-Two Method: TS(2)""; ""3.2.1 Commentary on the Construction""; ""3.3 An Order-p Method: TS(p)""; ""3.4 Convergence""; ""3.5 Application to Systems""; ""3.6 Postscript""
""4 Linear Multistep Methods�I: Construction and Consistency""""4.1 Introduction""; ""4.1.1 The Trapezoidal Rule""; ""4.1.2 The 2-step Adams�Bashforth method: AB(2)""; ""4.2 Two-Step Methods""; ""4.2.1 Consistency""; ""4.2.2 Construction""; ""4.3 k-Step Methods""; ""5 Linear Multistep Methods�II: Convergence and Zero-Stability""; ""5.1 Convergence and Zero-Stability""; ""5.2 Classic Families of LMMs""; ""5.3 Analysis of Errors: From Local to Global""; ""5.4 Interpreting the Truncation Error""; ""6 Linear Multistep Methods�III: Absolute Stability""
""6.1 Absolute Stability�Motivation""""6.2 Absolute Stability""; ""6.3 The Boundary Locus Method""; ""6.4 A-stability""; ""7 Linear Multistep Methods�IV: Systems of ODEs""; ""7.1 Absolute Stability for Systems""; ""7.2 Stiff Systems""; ""7.3 Oscillatory Systems""; ""7.4 Postscript""; ""8 Linear Multistep Methods�V: Solving Implicit Methods""; ""8.1 Introduction""; ""8.2 Fixed-Point Iteration""; ""8.3 Predictor-Corrector Methods""; ""8.4 The Newton�Raphson Method""; ""8.5 Postscript""; ""9 Runge�Kutta Method�I: Order Conditions""; ""9.1 Introductory Examples""
""9.2 General RK Methods""""9.3 One-Stage Methods""; ""9.4 Two-Stage Methods""; ""9.5 Three�Stage Methods""; ""9.6 Four-Stage Methods""; ""9.7 Attainable Order of RK Methods""; ""9.8 Postscript""; ""10 Runge-Kutta Methods�II Absolute Stability""; ""10.1 Absolute Stability of RK Methods""; ""10.1.1 s-Stage Methods of Order s""; ""Stage Methods of Order""; ""10.2 RK Methods for Systems""; ""10.3 Absolute Stability for Systems""; ""11 Adaptive Step Size Selection""; ""11.1 Taylor Series Methods""; ""11.2 One-Step Linear Multistep Methods""; ""11.3 Runge�Kutta Methods""
""11.4 Postscript""""12 Long-Term Dynamics""; ""12.1 The Continuous Case""; ""12.2 The Discrete Case""; ""13 Modified Equations""; ""13.1 Introduction""; ""13.2 One-Step Methods""; ""13.3 A Two-Step Method""; ""13.4 Postscript""; ""14 Geometric Integration Part I�Invariants""; ""14.1 Introduction""; ""14.2 Linear Invariants""; ""14.3 Quadratic Invariants""; ""14.4 Modified Equations and Invariants""; ""14.5 Discussion""; ""15 Geometric Integration Part II�Hamiltonian Dynamics""; ""15.1 Symplectic Maps""; ""15.2 Hamiltonian ODEs""; ""15.3 Approximating Hamiltonian ODEs""
""15.4 Modified Equations""
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