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

78 CHAPTER 4. IMPROVING ON EULER’S METHOD
Figure 4.3: Numerical solution of equation 4.80 for h = 0.190 (red), h = 0.205
(blue), and h = 0.23 (green).
.
Re(hλ) ≤ 0) then the method is said to be A-Stable. A-Stable methods are
preferable for stiff problems. A stronger requirement is Stiff Decay: consider
the generalized test equation
y ′ = λ(y − g(t)) (4.85)
where g(t) is any arbitrary, bounded function. Then we say that equation 4.85 has
stiff decay if for any fixed t n > 0,
lim |y n (t) − g(t n )| = 0. (4.86)
hRe(λ)→−∞
To determine the shape of this region we observe that in the complex plane we
can write
hλ = x + iy (4.87)
and therefore
1 ≥ |1 + x + iy| 2 (4.88)
= (1 + x + iy)(1 + x − iy) 2 (4.89)
= (1 + x) 2 + y 2 (4.90)
which is a disk of radius 1 centered at the point (−1, 0).
If λ ∈ R then this condition gives
−1 ≤ 1 + hλ ≤ 1 (4.91)
If λ < 0 then
−2 ≤ hλ ≤ 0 (4.92)
h ≤ − 2 λ
(4.93)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007