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

102 CHAPTER 5. RUNGE-KUTTA METHODS
and Kutta’s third-order formula
0 0
1/2 1/2 0
1 −1 2 0
1/6 2/3 1/6
(5.123)
The classical Runge-Kutta method is the four-stage fourth-order method
we have already met:
0 0
1/2 1/2 0
1/2 0 1/2 0
(5.124)
1 0 0 1 0
1/6 1/3 1/3 1/6
The region of absolute stability for all ERK methods will always be bounded,
and hence none of them can be A-stable. This is because the test equation will
always produce
y n = P (z)y n−1 (5.125)
where P (z) is a polynomial. There will always be sufficiently large values of z so that
|P (z) > 1. Since none of these methods are A-stable, they are generally not good
for stiff problems, as we have already seen for the Forward Euler Method (which is
a one-stage IRK). Thus one commonly uses implicit RK methods (IRK) instead.
5.5 Order Conditions
Explicit Runge-Kutta (ERK) methods have an order that is at most equal to the
number of stages; we have seen two, three, and four stage methods that have orders
equal to the number of stages. However, for n > 4, it is not possible to find an ERK
with an order that is equal to the number of stages. The best one can do is the
following.
Number of Stages 1 2 3 4-5 6 7-8 9-10 11 12 13-17
Best Possible Order 1 2 3 4 5 6 7 8 9 10
Unfortunately the proof of this statement is tedious - there is an explosion in the
number of terms that must be compared as the Taylor order is increased. Interested
students are referred to [7] for details.
To determine the error in a carrying out a single step of a Runge-Kutta method,
one must compare the successive terms in a Taylor series expansion of the exact and
computed solutions. This calculation is quite tedious: “The reader is no asked to
take a deep breath, take five sheets of reversed computer paper, remember the basic
rules of differential calculus, and begin the following computations...” (see [10, page
144]). The result is a set of order conditions that are necessary if a method is to
have a given order; however, these conditions are not, in general, sufficient.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007