Ordinary Differential Equations

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Ordinary Differential Equations

• Jyun-Ming Chen

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Contents

• Review
• Eulers method
• 2nd order methods
• Midpoint
• Heuns
• Runge-Kutta Method

• Systems of ODE
• Stability Issue

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Review

• DE (Differential Equation)
• An equation specifying the relations among the
  rate change (derivatives) of variables
• ODE (Ordinary DE) vs. PDE (Partial DE)
• The number of independent variables involved

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Review (cont)

• Solution of DE vs. Solution of Equation

• Solution of an equation

• Geometrically,

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Review (cont)
Need additional conditions to specify a solution

• Solution of an differential equation

• Geometrically

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Review (cont)

• Order of an ODE
• The highest derivative in the equation
• nth order ODE requires n conditions to specify
  the solution
• IVP (initial value problem) All conditions
  specified at the same (initial) point
• BVP (boundary value problem) otherwise

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IVP VS. BVPRevisit Shooting Problem
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IVP vs. BVP
Physical meaning
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Review (cont)

• Linearity
• No product nor nonlinear functions of y and its
  derivatives
• nth order linear ODE

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Focus of This Chapter

• Solve IVP of nth order ODE numerically
• e.g.,

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ODE (IVP)

• First order ODE (canonical form)
• Every nth order ODE can be converted to n first
  order ODEs in the following method

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Example
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End of Review
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The Canonical Problem
This is Eulers method
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Example
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Example (cont)
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Error Analysis(Geometric Interpretation)
Think in terms of Taylors expansion
If the true solution were a straight line,
then Euler is exact
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Error Analysis(From Taylors Expansion)
Eulers
Eulers truncation error O(Dx2) per step
1st order method
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Cumulative Error
y
x
Remark Dx? Error ? But computation time?
x 0
x T
Number of steps T/Dx Cumulative Err. (T/Dx) ?
O(Dx2) O(Dx)
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Example (Eulers)
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Methods to Improve Euler

• Motivated by Geometric Interpretation

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Midpoint Method
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Example (Midpoint)
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Heuns Method
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Example (Heuns)
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Remark

• Comparison of Euler, Heun, midpoint
• 1st order Euler
• 2nd order Heun, midpoint
• order
• All are special cases of RK (Runge-Kutta) methods

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RK Methods
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RK Methods (cont)
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Taylors Expansion
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RK 1st Order
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RK 2nd Order
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RK 2nd Order (cont)
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RK 4th Order

• Mostly commonly used one
• Higher order more evaluation, but less gain on
  accuracy

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System of ODE

• Convert higher order ODE to 1st order ODEs
• All methods equally apply, in vector form

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Example (Mass-Spring-Damper System)

• Governing Equation
• After setting the initial conditions x(0) and
  x(0), compute the position and velocity of the
  mass for any t gt 0

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Example (cont)
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Example (cont)
set Dt0.1
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Stability Symptom
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Stability (cont)

• Example Problem

Conditionally stable
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Discussion

• Different algorithm different stability limit
• Check Midpoint Method
• Different problem different stability limit
• use the previous problem as benchmark

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Implicit Method (Backward Euler)
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Example

• Remark
• Always stable (for this problem)
• Truncation error the same as Euler (only improve
  the stability)

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Linear System of ODE with Constant Coefficients
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Semi-Implicit Euler

• Not guaranteed to be stable, but usually is

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Stiff Set of ODE
Use the change of variable
Get the following solution
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Adaptive Stepsize

• Solving ODE numerically tracing the integral
  curve y(x)
• whats wrong with uniform step size
• Uniformly small waste effort
• Uniformly large might miss details

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Step Doubling

• Idea
• Estimate the truncation error by taking each step
  twice one full step, two half steps
• control the step size such that the estimated
  error is not too big.

1?(2h1)
2?(h1)
Desired h0
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Ex RK 2nd Order
Overhead of f(x,y) evaluations 2?42 6
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Example
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Ex (Semi-implicit Euler)
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