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

100 CHAPTER 5. RUNGE-KUTTA METHODS
(5.98)
different Runge-Kutta methods; individual ones are usually described by presenting
their Butcher Array,
c 1 a 11 a 12 · · · a 1s
c 2 a 21 a 22 · · · a 2s
. .
.
b 1 b 2 · · · b s
c s a s1 a s2 · · · a ss
The c i are the row-sums of the matrix. For explicit methods, the Butcher array is
strictly lower triangular. Thus it is common to omit the upper-diagonal terms when
writing the Butcher array.
An equivalent formulation to equations 5.95 and 5.96 is given by
Y i = y n−1 + h
y n = y n−1 + h
s∑
a ij f(t n−1 + c j h, Y j ) (5.99)
j=1
s∑
b i f(t n−1 + c i h, Y i ) (5.100)
i=1
Example 5.2. Show that Euler’s method is specified by the following Butcher array:
0 0
1
(5.101)
Solution. We have s = 1, a 11 = 0, c 1 = 0, and b 1 =0. Hence
K 1 = f(t n−1 + c 1 h, y n−1 + ha 11 K 1 ) (5.102)
= f(t n−1 , y n−1 ) (5.103)
y n = y n−1 + hb 1 K 1 (5.104)
= y n−1 + hf(t n−1 , y n−1 ) (5.105)
which reproduces Euler’s method.
=⇒
The most general two-stage explicit method takes the form
Since the c i are row sums this simplifies to
(5.106)
c 1 0
c 2 a 21 0
b 1 b 2
(5.107)
0 0
a a 0
b 1 b 2
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007