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Numerical Methods for Differential Equations

   Added On :  03.06.2016 02:18 pm
   Author :  Ya Yan Lu
   For :  Numerical Methods

Numerical Methods for Differential Equations by YA YAN LU.


Contents

1 ODE IVP: Explicit One-step Methods 4
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Euler and Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Local truncation error and order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Embedded Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 ODE IVP: Implicit One-step Methods 17
2.1 Stiff equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Implicit one-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 ODE IVP: Multi-step Methods 23
3.1 Explicit multi-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Implicit multi-step methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 ODE IVP: Stability Concepts 29
4.1 Zero stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Absolute stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 ODE Boundary Value Problems 34
5.1 The shooting method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Finite Difference Methods for Parabolic PDEs 42
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.2 Classical explicit method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Crank-Nicolson method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.4 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.5 Alternating direction implicit method . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7 Finite Difference Methods for Hyperbolic PDEs 52
7.1 First order hyperbolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2 Explicit methods for wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.3 Maxwell?s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 Finite Difference Methods for Elliptic PDEs 60
8.1 Finite difference method for Poisson equation . . . . . . . . . . . . . . . . . . . . . . 60
8.2 Fast Poisson solver based on FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.3 Classical iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.4 Conjugate gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.4.1 1-D optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.4.2 Subspace minimization problem . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.4.3 Orthogonal residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.4.4 The next conjugate direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.4.5 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.4.6 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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