The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 5. RUNGE-KUTTA METHODS 97
Figure 5.3: Region of absolute stability for the “classical” 4-stage Runge-Kutta
method.
Using the trigonometric identity 2 cos θ = e iθ + e −iθ ,
1 ≥ 2 (
24r 4 cos 4θ + (24)(4 + r 2 ) cos 3θ+
576
(5.72)
12(24 + 8r 2 + r 4 ) cos 2θ+ (5.73)
4(144 + 72r 2 + 12r 4 + r 6 ) cos θ+ (5.74)
(576 + 576r 2 + 144r 4 + 16r 6 + r 8 ) ) (5.75)
The region of absolute stability is illustrated in figure 5.3.
FIgure 5.4 compares the absolute error (y n (1) − e) at t = 1 using the 4-stage
Runge-Kutta and Euler’s method. Clearly the RK4 method, which is fourth order,
has much lower error and the error improves must faster with small step size then
does Euler.
Example 5.1. Compute the solution to the test equation y ′ = y, y(0) = 1 on [0, 1]
using the 4-stage Runge Kutta method with h = 1/2.
Solution. Since we start at t = 0 and need to compute through t = 1 we have to
compute two iterations of RK. For the first iteration,
k 1 = y 0 = 1 (5.76)
k 2 = y 0 + h 2 f(t 0, k 1 ) (5.77)
= 1 + (0.25)(1) = 1.25 (5.78)
k 3 = y 0 + h 2 f(t 1/2, k 2 ) (5.79)
= 1 + (0.25)(1.25) = 1.3125 (5.80)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge