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

CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 187
9.7 Runge-Kutta Methods for DAEs
Runge-Kutta methods have the advantage that it is possible to construct higherorder
A-stable methods for differential-algebraic equations. Furthermore, they are
useful for problems that are highly discontinuous because multistep methods such
as BDF need to be completely restarted after each discontinuity and are not completely
accurate for the first few iterations. In fact RK methods are sometimes used
as starter methods for these multistep methods. On the other hand, RK methods
have multiple function evaluations, which means they can be inefficient to implement:
a completely general s-stage method requires s 2 evaluations at each mesh
point. Furthermore, implicit Runge-Kutta methods suffer from order reduction, a
phenomenon in which the order of the method for the algebraic constraint is lower
than the order of the method for the differential system. We will only study RK
methods applied to the simplest system, namely, index-1 linear systems, and will
generalize our results without proof to more complicated systems. 6
Recall that we wrote the general Runge-Kutta method (equation 5.99) for the
initial value problem y ′ = f(t, y), y(t 0 ) = y 0 as
s∑
Y i = y n−1 + h a ij f(t n−1 + c j h, Y j ) (9.196)
y n = y n−1 + h
j=1
s∑
b i f(t n−1 + c i h, Y i ) (9.197)
This can be reformulated by replacing f(t n−1 + c j h, Y j ) with Y ′
j ,
i=1
Y i = y n−1 + h
y n = y n−1 + h
s∑
j=1
s∑
i=1
a ij Y ′
j (9.198)
b i Y ′
i (9.199)
where Y ′
j represents the numerical estimate of Y ′ at t n−1 +c j h. Then the discretization
of
F(t, y, y ′ ) = 0 (9.200)
becomes
⎛
F ⎝t n−1 + c i h, y n−1 + h
y n = y n−1 + h
⎞
s∑
a ij Y j, ′ Y i
′ ⎠ = 0 (9.201)
j=1
s∑
b i Y i ′ (9.202)
6 See [5][Chapter 4] for a more complete discussion. The material in this section roughly follows
the paper by L.R. Petzold (1986) “Order results for implicit Runge-Kutta methods applied to
differential/algebraic systems,” SIAM Journal of Numerical Analysis, 23(4):837-852. For details on
the proofs that are sketched and further examples the reader should refer to Petzold’s paper.
i=1
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge