CSE245:Lec4

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CSE245Lec4

• 02/24/2003

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Integration Method
Problem formulation
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Integration Method
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Integration Method
An example
(1)
(2)
From (1) ?
?
?
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Integration Method
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Integration Method
(3)
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Adams-Bashforth formula
?0 0
The first order Adams-Bashforth formula (forward
Euler)
The second order Adams-Bashforth formula
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Adams-Moulton formula
?0 ?0
The first order Adams-Moulton formula (backward
Euler)
The second order Adams-Moulton formula
(trapezoidal)
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Properties
Definition 1 Consistency
If MT/h is the number of steps, Global Error can
be estimated by
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Properties
Definition 2 Convergence
Definition 3 Stability
If ?h0k??, for any two initial condition x0, y0
and hT/Mlth0
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Stabilities
Backward Euler
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Stabilities
Froward Euler
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Stabilities
Trapezoidal
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Stabilities
A General Integration Equation
(4)
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Stabilities
Assume
(4)
stable
stable
Mi1, stable
Migt1, unstable
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A-stable

• Definition a method is A-stable, if the region
  of absolute stability includes the entire
  left-hand plane.
• Dahlquists theorem
• An A-stable method cannot exceed 2nd order
  accuracy.
• The most accurate A-stable method (smallest LTE)
  is the trapezoidal formula.

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Non-linear Circuit
Newton-Ralphson
(5)
(6)
(6)(5)
(7)
Newton-Ralphson iteration
(8)
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Non-linear Circuit
Newton-Ralphson
(8)(7)
(9)
Rewrite (4)
(10)
Compare (9) with (10)
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Oscillation
Reason1 x(0) is not close to x.
Reason2 error in derivative cause oscillation.
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Oscillation
Implications

1. Model equations must be continuous, and
   continuous derivatives.
2. Watch out the floating nodes.
3. Provide good initial guess for x(0).
4. Make sure , has ? greater than model error.