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# Numerical Solution of Ordinary Differential Equations

Added On :  03.06.2016 02:20 pm
Author :  E. Suli.
For :  Numerical Methods

Numerical Solution of Ordinary Differential Equations By E. Suli.

Contents
1 Picard?s theorem 1
2 One-step methods 4
2.1 Euler?s method and its relatives: the θ-method . . . . . . . . . . . . . . . . . . . . 4
2.2 Error analysis of the θ-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 General explicit one-step method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Runge?Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Absolute stability of Runge?Kutta methods . . . . . . . . . . . . . . . . . . . . . . 19
3 Linear multi-step methods 21
3.1 Construction of linear multi-step methods . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Zero-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Necessary conditions for convergence . . . . . . . . . . . . . . . . . . . . . . 30
3.4.2 Sufficient conditions for convergence . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Maximum order of a zero-stable linear multi-step method . . . . . . . . . . . . . . 37
3.6 Absolute stability of linear multistep methods . . . . . . . . . . . . . . . . . . . . . 43
3.7 General methods for locating the interval of absolute stability . . . . . . . . . . . . 45
3.7.1 The Schur criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7.2 The Routh?Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8 Predictor-corrector methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8.1 Absolute stability of predictor-corrector methods . . . . . . . . . . . . . . . 50
3.8.2 The accuracy of predictor-corrector methods . . . . . . . . . . . . . . . . . 52
4 Stiff problems 53
4.1 Stability of numerical methods for stiff systems . . . . . . . . . . . . . . . . . . . . 54
4.2 Backward differentiation methods for stiff systems . . . . . . . . . . . . . . . . . . 57
4.3 Gear?s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Nonlinear Stability 59
6 Boundary value problems 62
6.1 Shooting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.1.1 The method of bisection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1.2 The Newton?Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.1 Linear boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2.2 Nonlinear boundary value problem . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Collocation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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