Lecture 34 Ordinary Differential Equations BVP

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Lecture 34 - Ordinary Differential Equations -
BVP

• CVEN 302
• November 28, 2001

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Systems of Ordinary Differential Equations - BVP

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

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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

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Nonlinear Shooting Based on Secant Method

• Nonlinear
• ODE

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MATLAB Example in Nonlinear Shooting Method

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

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Nonlinear Shooting - Secant Method
y(x)
y(x)
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Nonlinear Shooting Based on Newtons Method

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

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Nonlinear Shooting Based on Newtons Method

• Nonlinear
• ODE-IVP
• Chain
• Rule

0
x and t are independent
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Nonlinear Shooting with Newtons Method

• Solve ODE-IVP
• Construct the auxiliary equations

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Nonlinear Shooting with Newtons Method

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

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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

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Finite-Difference Method

• General Two-Point BVPs
• Replace the derivatives by appropriate
  finite-difference approximations

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Finite-Difference Method

• Central difference approximations
• Tridiagonal system

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Finite-Difference Method

• Central Difference gt Tridiagonal system

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Finite-Difference Method for Nonlinear BVPs

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

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Finite-Difference Method for Nonlinear BVPs

• SOR method
• Iterative solution
• Convergence criterion

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Example 14.12 - MATLAB

• Note error in Text

fi negative sign
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Chapter 15

• Partial Differential Equations

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Classification of PDEs

• General form of linear second-order PDEs with two
  independent variables
• linear PDEs a, b, c,.,g f(x,y) only

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Heat Equation Parabolic PDE

• Heat transfer in a one-dimensional rod

x 0
x a
g1(t)
g2(t)
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Discretize the solution domain in space and time
with h ?x and k ?t
t
Time (j index)
x
space (i index)
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Initial and Boundary Conditions
Explicit Euler method
u(a, t) g2(t)
u(0, t) g1(t)
Initial conditions u(x,0) f(x)
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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
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Explicit Method

• Explicit Euler method for heat equation
• Rearrange

Stability
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Explicit Euler Method

• Stable
• Unstable (negative coefficients)

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Heat Equation Explicit Euler Method
r 0.5
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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
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Example

• Explicit Euler method
• First step t 0.05

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• 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
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Heat Equation Time-dependent BCs
r 0.4
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Numerical Stability

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

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Explicit Euler Method Stability
r 1
Unstable !!
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Implicit Euler method
Unconditionally Stable
u(a, t) g2(t)
u(0, t) g1(t)
Initial conditions u(x,0) f(x)
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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
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Implicit Euler Method

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

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Implicit Euler Method
r 2
Unconditionally stable
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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
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Example

• Implicit Euler method

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• Solve the tridiagonal matrix

28.96
38.51
46.19
1
60e -2t
20e -t
0
1
2
3
4
0
20 40 x
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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)
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Crank-Nicolson Method

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

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General Two-Level Method

• General two-stage method for heat equation
• Weighted-average of spatial derivatives between
  two time levels n and n1

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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
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Example

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

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• Solve the tridiagonal matrix

29.42
39.30
47.43
1
60e -2t
20e -t
0
1
2
3
4
0
20 40 x
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Implicit Euler method
r 2
Unconditionally stable
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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
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Insulated Boundary

• No heat flux at x a

ux(a,t)0
xn1
xn-1
xn
x a