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

184 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS
There is no solution for (x n , y n ) in terms of the variables at earlier times. This can
be seen be rewriting the system as
x n − t n y n = h(x n−1 − t n y n−1 ) (9.171)
x n − t n y n = f(t n ) (9.172)
or in matrix form
( ) ( )
1 −tn xn
=
1 −t n y n
( )
h(xn−1 − t n y n−1 )
f(t n )
The matrix of coefficients is singular hence there is no solution.
(9.173)
9.6 BDF Methods for DAEs
While the numerical solution of initial value problems for ordinary differential equations
has a long history going back over one hundred years, the numerical solution of
DAEs is mostly still an open research issue (see [5] for the more common methods)
with significant interest only dating back to the 1970’s. Since many of the scientific
problems of interest turn out to be stiff, early research focused first on BDF methods
and later on some RK methods that are known to be better for stiff problems.
The first DAE algorithms published were BDF methods. 5 The solution of linear
and index-1 systems are now understood but methods for higher index systems still
form the basis of much current methods.
In general variable step sizes can cause problems with DAEs. For example, it is
not possible to solve an index-3 system using implicit Euler with variable step-size
because of the following theorem because it will have O(1) errors.
Theorem 9.11. The global error for a k-step BDF applied to an index-ν system is
O(h n ) where n = min(k, k − ν + 2)
Furthermore, all multistep (as well as Runge-Kutta) methods are unstable for
higher index (≥ 3) systems, and in some cases fail for specific index-1 and index-2
systems. Some convergence results for semi-implicit and fully-implicit index-1 systems,
semi-explicit index-2 systems, and index-3 systems in Hessenberg form have
been worked out.
We can write the general k-step BDF method for a scalar ODE (see 6.105) as
k∑
a j y k−j = hb 0 Dy n (9.174)
j=0
where D is the differential operator Dy = y ′ . Then the general BDF method
becomes
0 = F (t n , y n , Dy n ) (9.175)
5 C.W.Gear (1971) “Simultaneous Numerical Solution of Differential-Algebraic Equations,” IEEE
Transactions on Circuit Theory, 18:89-95
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007