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5.4 Runge-Kutta 4 Method. Carl Runge (1856-1927). Martin Wilhelm Kutta (1 867-1944 ). Motivation. With Euler’s method, error is described by a straight line: i.e., it is proportionate to (linear in) D t . We say that error is O( D t ) : “Order D t”, or “Big-O D t ” Can we do better?. - PowerPoint PPT Presentation
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5.4 Runge-Kutta 4 Method
Carl Runge(1856-1927)
Martin Wilhelm Kutta
(1867-1944)
Motivation• With Euler’s method, error is described by a straight line: i.e., it is proportionate to
(linear in) t.• We say that error is O(t) : “Order t”, or “Big-O t”• Can we do better?
First Estimate ∂1• Recall update rule from Euler’s 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
∆t = 8
∂1 = 80
Second Estimate ∂2
• Second estimate ∂2 uses the halfway point along the
line segment to ∂1
Second Estimate ∂2• Combined with slope 0.10 (from dP/dt = 0.10P), this gives us a new
endpoint = (0.1)(140)(8) = 112, which is the second estimate ∂2.
• ∂2 = f(tn-1+0.5t, Pn-1+0.5∂1) t
∆t = 8
∂2= 112
Third Estimate ∂3
• Third estimate ∂3 uses the halfway point along the line segment from ∂2
• ∂3 =(0.1)(156)(8) = 124.8
Third Estimate ∂3
• (0.1)(156)(8) = 124.8, which is the third estimate ∂3.
• ∂3 = f(tn-1+0.5t, Pn-1+0.5∂2) t
∂3= 124.8
Fourth Estimate ∂4
• Fourth estimate ∂4 is taken at end of interval
• ∂4 =(0.1)(224.8)(8) = 179.84
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
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 + 2*112 + 2*124.8 + 179.84) / 6
= 222.24
• Relative error = |222.4 - 222.55| / |222.55| = 0.14%(Compare 19% for Euler’s Method)
• RK4 error is O(t4) : a very small number, because error is < 1.
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-1+0.5t, Pn-1+0.5∂1) t∂3 = f(tn-1+0.5t, Pn-1+0.5∂2) t∂4 = f(tn-1+ t, Pn-1+∂3) t
Runge-Kutta 4 in Excel
Runge-Kutta 4 in Excel
Runge-Kutta 4 in Excel
Runge-Kutta 4 in Excel
Runge-Kutta 4 in Excel
Runge-Kutta 4 in Excel