Multistep methods

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• Multistep methods
• previous methods use information at xi to
  predict yi1
• multistep methods use information from xi-1,
  xi-2, etc.
• leads to better results

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Simple 2nd order method to start Look at Heun
method Euler predictor - O(h2)
Trapezoid corrector - O(h3)
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To improve predictor, use third order Euler
along with same old corrector
iterated until converged
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Can make one more modification - a predictor
modifier Comes from error analysis of yi10
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• So sequence is
• predict
• adjust prediction
• correct
• convereged?
• if not, correct again

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Example of non-self starting Heun method Need two
initial conditions
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• Non-self starting Heun method is O(h3) accurate
  in predictor and corrector
• Uses
• midpoint method for predictor (open)
• trapezoid method for corrector (closed)
• Try higher order methods

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• We have a number of options
• Newton-Cotes formulas
• open
• closed
• Adams formulas
• open (Adams-Bashforth)
• closed (Adams-Moulton)

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Newton-Cotes approach - Open methods Given can
write
n is order of interpolating polynomial
10
n1
(midpoint method)
n2
n3
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Newton-Cotes -- Closed methods General form
Integral evaluated using closed Newton-Cotes
formulas
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n1
(trapezoid rule)
n2
(Simpsons 1/3 rule)
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Adams formulas Open formulas - Adams
Bashforth Example How to get 2nd order
Adams-Bashforth
Taylor series
rewrite
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Use backwards difference for derivative
and substitute
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End up with second order Adams-Bashforth
with error term
Can do the same proceedure on other order Taylor
series, other finite differences Get higher order
Adams Bashforth
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General form for Adams-Bashforth
The bs are listed in your book Example 4th order