Lecture Notes in Differential Equations - Bruce E. Shapiro

372 LESSON 33. NUMERICAL METHODS
k 4 = y 1 + hf(t 1.5 , k 3 ) (33.98)
= 1.64844 + (0.5)(2.16358) (33.99)
= 2.73023 (33.100)
y 2 = y 1 + h 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (33.101)
= 1.64844 + .5 1.64844 + 2(2.06055) + 2(2.16358) + 2.73023
6
(33.102)
= 2.71735 (33.103)
This gives us a numerical approximation of y(1) ≈ 2.71735, and error of
approximately 0.034% (the exact value is e ≈ 2.71828. By comparison, a
forward Euler computation with the same step size will yield a numerical
result of 2.25, an error approximately 17%.
Since it is an explicit method, the Runge-Kutta 4-stage method is very easy
to implement in a computer, even though calculations are very tedious to
do by hand