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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 189
we can transform to canonical form
where
( ) I 0
Ỹ i ′ +
0 N
Partitioning the solutions as
( ) ⎛ C 0
⎝ỹ
0 I n−1 + h
⎞
s∑
a ij Ỹ j
′ ⎠ = g(t n−1 + c i h) (9.211)
j=1
ỹ n = ỹ n−1 + h
s∑
b i Ỹ i ′ (9.212)
i=1
Ỹ ′ i = Q −1 Y ′ i, ỹ n = Q −1 y n , g(t) = P y(t) (9.213)
Ỹ =
( ( ( U u p
, ỹ = , ˜g =
V)
v)
q)
(9.214)
and multiplying out the matrix expression in equation 9.211,
⎛
⎞
s∑
U ′ i + C ⎝u n−1 + h a ij U ′ ⎠
j = p(t n−1 + c i h) (9.215)
j=1
NV ′ i + v n−1 + h
s∑
a ij V j ′ = q(t n−1 + c i h) (9.216)
j=1
Equation 9.215 is an ordinary differential equation that has been discretized using
an s-stage Runge-Kutta method, which has properties that we have already studied.
Equation 9.216 contains the “singular” part of the system, and because the differential
and singular parts of the system are completely decoupled, it suffices to just
study the canonical singular subsystem.
Let v n,p , q p denote the p th component of the corresponding vectors v n , q, let V j,p
be the j th stage derivative for the p th component of V j , and denote the RK matrix
(a ij ) by A. Then for the index-1 system (with N = 0) equation 9.216 becomes
v n−1,p + hAV ′ = q p (t n−1 + c i h) (9.217)
where V is the vector of stage derivatives corresponding to component p. Equation
9.217 must hold for all values of h, and in the limit h → 0,
v n−1,p = q p (t n−1 ) (9.218)
Omitting the index p to simplify the notation and assuming that the matrix A is
nonsingular,
V ′ = 1 h A−1 [(q(t n−1 + c i h) − v n−1 )] i=1,...,s (9.219)
= 1 h A−1 [(q(t n−1 + c i h) − q(t n−1 ))] i=1,...,s (9.220)
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge