Вычислительная математика - ИСЭМ СО РАН

emains small. Dahlquist [3] investigated the special stability problem connected with stiff equations.
He introduced the concept of A-stability, and we quote the definition as the following:
Definition 1. The stability region R associated with a multistep formula is defined as the set
R = {hλ : the formula applied to y ′ = λy, y(x 0 ) = y 0 , with constant stepsize h > 0, produce a sequence
(y n ) satisfying that y n → 0as n → ∞}.
Definition 2. A formula is A-stable if the stability region associated with that formula contains
the open left half place.
Definition 3. [4] A convergent linear multistep method isA(α)-stable, 0 < α < π/2, if S ⊃
S α = {µ : |arg(−µ)| < α, µ ≠ 0}. A method is A(0)-stable if it is A(α)-stable for some (sufficiently
small)α > 0.
We apply formulas Eqs. (3), (7), and (8) to y ′ = λy,y(x 0 ) = y 0 , and manipulation is skipped here.
Let µ = λh, we have the following results:
a. For Equation (5), y 2m = R(µ) m y 0 , y 2m+1 = S(µ)R(µ) m y 0 , where y m+1 = 6−µ2 y
6−6µ+2µ 2 m ≡ S(µ)y n
and y m+2 = 3+3µ+µ2 y
3−3µ+µ 2 m ≡ R(µ)y n . We have plotted the graph of the stability region, and given in the
following, where the region is the intersection of two regions, one is the red part and the other is outside
the blue part. It shows that the formula (5) is an A-stability formula. By the similar approaches, we
can also plot the regions of formula (7), and (8).
b. Formula (7), (8) respectively:
Figure 1: Stability region of Formula 3.1.
Figure 2: Stability regions of Formula 3.1 and 3.2.
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