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Lecture 34 - Ordinary Differential Equations -

BVP

- CVEN 302
- November 28, 2001

Systems of Ordinary Differential Equations - BVP

- Shooting Method for Nonlinear BVP
- Finite Difference Method
- Partial Differential Equations

Shooting Method for Nonlinear ODE-BVPs

- Nonlinear
- ODE
- Consider

with guessed slope t - Use the difference between u(b) and yb to adjust

u(a) - m(t) u(b, t) - yb is a function of the guessed

value t - Use secant method or Newton method to find the

correct t value with m(t) 0

Nonlinear Shooting Based on Secant Method

- Nonlinear
- ODE

MATLAB Example in Nonlinear Shooting Method

- Nonlinear shooting with secant method
- Convert to two first-order ODE-IVPs
- Update t using the secant method

Nonlinear Shooting - Secant Method

y(x)

y(x)

Nonlinear Shooting Based on Newtons Method

- Nonlinear
- ODE
- Check for convergence of m(t)

Nonlinear Shooting Based on Newtons Method

- Nonlinear
- ODE-IVP
- Chain
- Rule

0

x and t are independent

Nonlinear Shooting with Newtons Method

- Solve ODE-IVP
- Construct the auxiliary equations

Nonlinear Shooting with Newtons Method

- Calculate m(t) -- deviation from the exact BC
- Update t by Newtons method

Finite-Difference Methods

- Divide the interval of interest into subintervals
- Replace the derivatives by appropriate

finite-difference approximations in Chapter 11 - Solve the system of algebraic equations by

methods in Chapters 3 and 4 - For nonlinear ODEs, methods in Chapter 5 may be

used

Finite-Difference Method

- General Two-Point BVPs
- Replace the derivatives by appropriate

finite-difference approximations

Finite-Difference Method

- Central difference approximations
- Tridiagonal system

Finite-Difference Method

- Central Difference gt Tridiagonal system

Finite-Difference Method for Nonlinear BVPs

- Nonlinear ODE-BVPs
- Evaluate fi by appropriate finite-difference

approximations

Finite-Difference Method for Nonlinear BVPs

- SOR method
- Iterative solution
- Convergence criterion

Example 14.12 - MATLAB

- Note error in Text

fi negative sign

Chapter 15

- Partial Differential Equations

Classification of PDEs

- General form of linear second-order PDEs with two

independent variables - linear PDEs a, b, c,.,g f(x,y) only

Heat Equation Parabolic PDE

- Heat transfer in a one-dimensional rod

x 0

x a

g1(t)

g2(t)

Discretize the solution domain in space and time

with h ?x and k ?t

t

Time (j index)

x

space (i index)

Initial and Boundary Conditions

Explicit Euler method

u(a, t) g2(t)

u(0, t) g1(t)

Initial conditions u(x,0) f(x)

Heat Equation

- Finite-difference

t

tj1

u(x,t)

(i,j1)

t

tj

(i,j)

(i1,j)

(i-1,j)

x

xi

xi1

xi-1

x

Forward-difference Central-difference at time

level j

Explicit Method

- Explicit Euler method for heat equation
- Rearrange

Stability

Explicit Euler Method

- Stable
- Unstable (negative coefficients)

Heat Equation Explicit Euler Method

r 0.5

Example Explicit Euler Method

- Heat Equation (Parabolic PDE)
- c 0.5, h 0.25, k 0.05

2

60e -2t

20e -t

1

0

1

2

3

4

0

20 40 x

Example

- Explicit Euler method
- First step t 0.05

- Second step t 0.10

29.61

40

47.72

60e -2t

20e -t

30

40

50

1

2

3

4

0

20 40 x

Heat Equation Time-dependent BCs

r 0.4

Numerical Stability

- Stability for Explicit Euler Method
- It can be shown by Von Neumann analysis that
- Switch to Implicit method to avoid instability

Explicit Euler Method Stability

r 1

Unstable !!

Implicit Euler method

Unconditionally Stable

u(a, t) g2(t)

u(0, t) g1(t)

Initial conditions u(x,0) f(x)

Implicit Method

- Finite-difference

t

(i1,j1)

(i-1,j1)

(i,j1)

tj1

T(x,t)

t

tj

(i,j)

x

xi

xi1

xi-1

x

Forward-difference Central-difference at time

level j1

Implicit Euler Method

- Implicit Euler method for heat equation
- Tridiagonal matrix (Thomas algorithm)
- Unconditionally stable

Implicit Euler Method

r 2

Unconditionally stable

Example Implicit Euler Method

- Heat Equation (Parabolic PDE)
- c 0.5, h 0.25, k 0.1

1

60e -2t

20e -t

0

1

2

3

4

0

20 40 x

Example

- Implicit Euler method

- Solve the tridiagonal matrix

28.96

38.51

46.19

1

60e -2t

20e -t

0

1

2

3

4

0

20 40 x

Crank-Nicolson method

Implicit Euler method first-order in

time Crank-Nicolson second-order in time

u(a, t) g2(t)

u(0, t) g1(t)

Initial conditions u(x,0) f(x)

Crank-Nicolson Method

- Crank-Nicolson method for heat equation
- Average between two time levels
- Tridiagonal matrix
- Unconditionally stable (neutrally stable)
- Oscillation may occur

General Two-Level Method

- General two-stage method for heat equation
- Weighted-average of spatial derivatives between

two time levels n and n1

Example Crank-Nicolson Method

- Heat Equation (Parabolic PDE)
- c 0.5, h 0.25, k 0.1

1

60e -2t

20e -t

0

1

2

3

4

0

20 40 x

Example

- Crank-Nicolson method
- Tridiagonal matrix (r 0.8)

- Solve the tridiagonal matrix

29.42

39.30

47.43

1

60e -2t

20e -t

0

1

2

3

4

0

20 40 x

Implicit Euler method

r 2

Unconditionally stable

Heat Equation with Insulated Boundary

- No heat flux at x 0 and x a

x 0

x a

ux(a,t) 0

ux(0,t) 0

Insulated Boundary

- No heat flux at x a

ux(a,t)0

xn1

xn-1

xn

x a

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