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

188 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
Definition 9.11. Let y n = y n−1 + hψ(t n−1 , y n−1 ) be a numerical method. Then we
define the local error as
d n = y ( t n−1 ) + hψ(t n−1 , y(t n−1 )) − y(t n ) (9.203)
The local error defined above is just h times the local truncation error we found earlier
- if a method has LTE O(h p ) then it has local error O(h p+1 ). (See definition 4.5.)
For the Runge-Kutta method 9.202,
ψ =
s∑
b i Y i ′ (9.204)
i=1
where the Y i ′ are determined by solving the system of algebraic equations 9.201. If
the equation is linear and solvable (hence with regular pencil)
then the RK method becomes
⎛
along with equation 9.202.
AY i ′ + B ⎝y n−1 + h
Ay ′ + By = f(t) (9.205)
⎞
s∑
a ij Y j
′ ⎠ = f(t n−1 + c i h) (9.206)
j=1
We have previously seen that when the linear system is solvable then there exist
matrices P and Q such that it can be transformed into canonical form with
( ) ( )
I 0
C 0
P AQ = ; P BQ =
(9.207)
0 N
0 I
where
N = diag(N 1 , . . . , N K ) (9.208)
N i = subdiagonal(1, . . . , 1) (9.209)
Here N is nilpotent of nilpotency k, where k is the index of if the DAE; hence for
index-1 systems, we have N = 0.
Definition 9.12. If P AQ = N and P BQ = I with N nilpotent then the system is
called completely singular.
Definition 9.13. If N has the form subdiagonal1, . . . , 1 for a completely singular
system then the system is called a canononical completely singular system.
Premultiplying the linear system by P gives
⎛
⎞
s∑
P AQQ −1 Y i ′ + P BQQ −1 ⎝y n−1 + h a ij Y j
′ ⎠ = P f(t n−1 + c i h) (9.210)
j=1
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007