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

186 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
For the BDF method 9.176,
Dy n = y ′ n + O(h k ) (9.188)
hence
D j v n = v (j)
n + O(h k ) (9.189)
where the superscript v (j) represents the j-th derivative, and therefore
∑ν−1
v n = (−1) j N j (h (j)
n (t) + O(h k )) (9.190)
j=0
Therefore the method converges with order h k .
For proofs of the following results see [5].
Theorem 9.13. Let F(t, y, y ′ ) be a fully-implicit index-1 system. Then its numerical
solution by the k-step (k < 7) BDF with fixed step-size converges to O(h k ) if all
initial values are correct to O(h k ) and the Newton iteration is solved at aech step to
O(h k+1 ) accuracy.
Theorem 9.14. Let
f(t, y, y ′ , z) = 0 (9.191)
g(t, y, z) = 0 (9.192)
be a solvable index-2 DAE such that f and g are continuously differentiable (as
many times as necessary), that ∂f/∂y ′ is defined and bounded, and that ∂g/∂y has
constant rank. Then the k-step (k < 7) BDF method applied to this DAE converges
with order O(h k ) if the initial values are all accurate to O(h k ) and the Newton
iteration is performed at each step to O(h k+1 ).
Theorem 9.15. Let
x ′ = F(t, x, y, z) (9.193)
y ′ = G(t, x, y) (9.194)
0 = H(t, y) (9.195)
be an index-three Hessenberg system. Then the k-step (k < 7) BDF with constant
step size converges to O(h k ) after k + 1 steps if the starting values are consistent to
O(h k+1 ) and the Newton iteration at each step is performed to O(h k+2 ) when k ≥ 2,
and to O(h k+3 ) when k = 1.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007