5'4 RungeKutta 4 Method

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5.4 Runge-Kutta 4 Method
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Carl Runge(1856-1927)
Martin Wilhelm Kutta(1867-1944)
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Motivation

• With Eulers method, error is described by a
  straight line i.e., it is proportionate to
  (linear in) Dt.
• We say that error is O(Dt) Order Dt, or
  Big-O Dt
• Can we do better?

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First Estimate ?1

• Recall update rule from Eulers Method
• Pn ? Pn-1 f(tn-1, Pn-1 ) ?t
• where f is the derivative function
• We call f(tn-1, Pn-1 ) 80 the first estimate,
  or ?1

?1 80
?t 8
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Second Estimate ?2

• Second estimate ?2 uses the halfway point along
  the line segment to ?1

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Second Estimate ?2

• Combined with slope 0.10 (from dP/dt 0.10P),
  his gives us a new endpoint (0.1)(140)(8)
  112, which is the second estimate ?2.
• ?2 f(tn-10.5?t, Pn-10.5?1) ?t

?2 112
?t 8
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Third Estimate ?3

• Third estimate ?3 uses the halfway point along
  the line segment from ?2
• ?3 (0.1)(156)(8) 124.8

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Third Estimate ?3

• (0.1)(156)(8) 124.8, which is the third
  estimate ?3.
• ?3 f(tn-10.5?t, Pn-10.5?2) ?t

?3 124.8
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Fourth Estimate ?4

• Fourth estimate ?4 is taken at end of interval
• ?4 (0.1)(224.8)(8) 179.84

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Fourth Estimate ?4

• ?4 (0.1)(224.8)(8) 179.84, which is the fourth
  estimate ?4.
• ?4 f(tn-1 ?t, Pn-1?3) ?t

?4 179.8
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Runge-Kutta 4 Estimate

• Bring it all together a weighted average that
  privileges the middle values
• Pn Pn-1 (?1 2?2 2?3 ?4) / 6
• 100 (80 2112 2124.8 179.84) /
  6
• 222.24
• Relative error 222.4 - 222.55 / 222.55
  0.14
• (Compare 19 for Eulers Method)
• RK4 error is O(Dt4) a very small number,
  because error is lt 1.

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Runge-Kutta 4 Algorithm
t ? t0 P(t0)? P0 Initialize NumberOfSteps for n
going from 1 to NumberOfSteps do the following
tn ? t0 n?t ?1 f(tn-1, Pn-1 ) ?2
f(tn-10.5?t, Pn-10.5?1) ?t ?3
f(tn-10.5?t, Pn-10.5?2) ?t ?4 f(tn-1 ?t,
Pn-1?3) ?t
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Runge-Kutta 4 in Excel
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Runge-Kutta 4 in Excel
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Runge-Kutta 4 in Excel
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Runge-Kutta 4 in Excel
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Runge-Kutta 4 in Excel
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Runge-Kutta 4 in Excel