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To determine the coefficients, k1 and k2 are expanded in a Taylor series about (tn, xn),
giving:
k1 = hn fn
k2 = hn
fn + bk1
¶ f
¶x
n
+ ahn
¶ f
¶t
n
+ . . .
(6.9)
Hence:
xn+1 = xn + (g1 + g2)hn fn + g2bh2
n
¶ f
¶x
n
fn + g2ah2
n
¶ f
¶t
n
+ . . . (6.10)
In these equations, fn has been used as a shorthand notation for f (xn, tn). The expansion
of xn+1 can be compared with the Taylor series for the actual local solution
zn(t):
zn(tn+1) = zn(tn) + hnz˙n(tn) + h2
n
2
¨zn(tn) + . . .
= xn + hn fn + h2
n
2
¶ f
¶x
n
fn +
¶ f
¶t
n
+ . . . (6.11)
Matching the coefficients of the powers of hn yields three equations involving the
unknown coefficients:
g1 + g2 = 1
g2b = 1
2 (6.12)
g1a = 12
It is now possible to choose one coefficient as parameter, which results in a oneparameter
family of Runge-Kutta methods. For instance, if a is chosen as a parameter,
we get:
k1 = hn f (xn, tn)
k2 = hn f (xn + ak1, tn + ahn) (6.13)
xn+1 = xn +
1 −
1
2a
k1 + 1
2a
k2
Obvious choices of a are 12
, which yields a method closely related to the rectangle
rule, and 1, yielding a method related to the trapezoid rule. The method is of second
order, because no choice of a can eliminate the h3 terms, which were neglected in the
above equations.
Higher-order Runge-Kutta methods can be derived in a similar manner. Although
these higher-order methods provide an improvement of accuracy, this comes at the
cost of additional derivative evaluations. For a given overall accuracy in a time response
calculation, there is a trade-off between many small steps with a lower-order
method, or fewer steps, but more derivative evaluations with a higher-order method.
6.1. Numerical integration methods 79
ai bij gi
gi
0 35
384
5179
57600
15
15
0 0
3
10
3
40
9
40
500
1113
7571
16695
45
44
45 −56
15
32
9
125
192
393
640
89
19372
6561 −25360
2187
64448
6561 −212
729 −2187
6784 − 92097
339200
1 9017
3168 −355
33
46732
5247
49
176 − 5103
18656
11
84
187
2100
1 35
384 0 500
1113
125
192 −2187
6784
11
84 0 1
40
Table 6.1: Coefficients of the Dormand-Prince (4,5) pair
In order to reduce the number of function evaluations, mathematicians have developed
several algorithms that combine Runge-Kutta integration with an estimation
of the error in the computed function at each time-step; the error estimate can be used
to control the step-size automatically in order to meet a specified accuracy. Over the
years, several so-called Runge-Kutta pairs have been developed. These pairs combine
two Runge-Kutta methods, usually of adjacent orders p q, which share their a and
b coefficients and differ only in the g’s.
For p = 1, . . . , 4 it is possible to obtain a pth-order method with p function evaluations,
while for p = 5 or 6, p function evaluations will produce a (p − 1)st-order
method only [17]. However, for Runge-Kutta pairs the ‘extra’ function evaluation is
not wasted, as the difference between the two solutions provides a convenient error
estimate. Such pairs are defined by:
ki = hn f (xn +
i−1
å
j=1
bij ki , tn + ai hn) (6.14)
The new value xn+1 is obtained using a weighted combination of the six k’s. The two
Runge-Kutta methods that form a pair result in two solutions for xn+1:
xn+1 = xn +
Nå
i=1
gi ki
x
n+1 = xn +
Nå
i=1
g
i ki (6.15)
where N is the number of function evaluations. For 4th/5th-order pairs, N 6. In
practice, the lower-order solution x
n+1 is not actually computed; instead the error
estimate:
dn xn+1 − x
n+1 =
Nå
i=1
(gi − g
i ) (6.16)
is determined and used for step-size control.
80 Chapter 6. Analytical tools
The coefficients ai, bij, and gi can again be found by by expanding the k’s in Taylor
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