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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 191
is given by
⎛
⎞
s∑
F ⎝t n−1 + c i h, z n−1 + h a ij Z ′ j + δ n (i) , Z ′ ⎠
i = 0 (9.232)
j=1
z n = z n−1 + h
s∑
i=1
b i Z ′ i + δ s+1
n (9.233)
where z 0 = y 0 + δ s+1
0 .
Definition 9.16. Let (z n , δ 0 , . . . , δ s+1 ) be a perturbation of the IRK method y n
for a particular differential algebraic equation F (t, y, y ′ ) = 0, with |δ n (i) | ≤ ∆, i =
1, . . . , s + 1, for some number ∆. Then if |z n − y n | ≤ K 0 ∆ for 0 < h ≤ h 0 , for some
constants K 0 and h 0 that depend only on the method and the differential algebraic
equation, then we say the method is strictly stable for the DAE.
Theorem 9.17. An IRK method with A nonsingular is strictly stable for a linear
constant-coefficient index-1 differential algebraic equation iff and only iff
where 1 is as defined in equation 5.211.
|1 − b T A −1 1| < 1 (9.234)
Recall from that the stability function R(z) = y n /y n−1 satisfies
Taking the limit as |z| → ∞,
R(z) = 1 + zb T (I − zA) −1 1 (9.235)
lim R(z) = 1 −
|z|→∞ bT A −1 1 (9.236)
From theorem 9.17 implies that an implicit Runge-Kutta method is stable when
r =
lim |R(z)| < 1 (9.237)
|z|→∞
Definition 9.17. k c is called the constant coefficient order of an implicit Runge-
Kutta method if the method converges with global error O(h kc ) for all linear constant
coefficient index-one systems.
Theorem 9.18. Let A, b, c be an implicit Runge-Kutta method that is strictly stable,
with constant-coefficent order k c , algebraic order k a , and differential order k d . Then
k c = min(k a + 1, k d ) (9.238)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge