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

90 CHAPTER 5. RUNGE-KUTTA METHODS
error of h p . We also observe that O(h) Taylor method is equivalent to Euler’s
method.
The second order Taylor method has
y n = y n−1 + hf + h2
2 (f t + f y f) (5.11)
For the test equation y ′ = λy we have f = λy, f t = 0, f y = λ, and hence
y n = y n−1 + hλy n + h2
2 λ2 y n (5.12)
)
=
(1 + hλ + (hλ)2 y n−1 (5.13)
2
Absolute stability occurs when
∣ y n ∣∣∣ (hλ)2
∣ =
y n−1
∣1 + hλ + 2 ∣ ≤ 1 (5.14)
To find the shape of this region, set z = hλ = re iθ , and square both sides of equation
5.14 using the identity |z| 2 = zz∗ to get
(1 + re iθ + r2 e 2iθ ) (1 + re −iθ + r2 e −2iθ )
≤ 1 (5.15)
2
2
1 + re −iθ + r2 e −2iθ
+ re iθ + r 2 + r 3 e −iθ r2 e 2iθ
2
2
( e −iθ + e iθ )
2r
+ r 2 e−2iθ + e 2iθ
+ r 3 e−iθ + e iθ
2
2
2
+ r 3 e iθ + r4
4 ≤ 1 (5.16)
+ r 2 + r4
4 ≤ 0 (5.17)
2r cos θ + r 2 cos 2θ + r 3 cos θ + r 2 + r4
4 ≤ 0 (5.18)
To plot this equation pick some small interval δθ and do the following:
• Set θ = 0
• Repeat
1. Set θ = θ + δθ
2. Solve 5.18 for r, accepting only real solutions
3. Plot the point (r, θ)
until θ = 2π
In Mathematica we can use Solve to find the solution of 5.18. For example, to find
r(π),
f[r , theta ] := 2r * Cos[theta] + r^2 * Cos[2*theta] + r^3 Cos[theta]
+ 0.25* r^4 + r^2;
Solve[f[r,Pi]==0]
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007