1. 6.4 Runge-Kutta 4 Method

2. Martin Wilhelm Kutta (1867-1944)
Carl Runge ( )
Martin Wilhelm Kutta ( )

3. 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?

4. 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
∂1 = 80
∆t = 8

5. Second Estimate ∂2
Second estimate ∂2 uses the halfway point along the line segment to ∂1

6. 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+0.5t, Pn-1+0.5∂1) t
∂2= 112
∆t = 8

7. Third Estimate ∂3
Third estimate ∂3 uses the halfway point along the line segment from ∂2
∂3 =(0.1)(156)(8) = 124.8

8. Third Estimate ∂3
(0.1)(156)(8) = 124.8, which is the third estimate ∂3.
∂3 = f(tn-1+0.5t, Pn-1+0.5∂2) t
∂3= 124.8

9. Fourth Estimate ∂4 ∂4 =(0.1)(224.8)(8) = 179.84
Fourth estimate ∂4 is taken at end of interval
∂4 =(0.1)(224.8)(8) =

10. Fourth Estimate ∂4
∂4 =(0.1)(224.8)(8) = , which is the fourth estimate ∂4.
∂4 = f(tn-1+ t, Pn-1+∂3) t
∂4= 179.8

11. 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
= (80 + 2* * ) / 6 =
Relative error = | | / |222.55| = 0.14%
(Compare 19% for Euler’s Method)
RK4 error is O(t4) : a very small number, because error is < 1.

12. Runge-Kutta 4 Algorithm
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+0.5t, Pn-1+0.5∂1) t
∂3 = f(tn-1+0.5t, Pn-1+0.5∂2) t
∂4 = f(tn-1+ t, Pn-1+∂3) t

13. Runge-Kutta 4 in Excel

14. Runge-Kutta 4 in Excel

15. Runge-Kutta 4 in Excel

16. Runge-Kutta 4 in Excel

17. Runge-Kutta 4 in Excel

18. Runge-Kutta 4 in Excel