1
COMPUTER SCIENCE 349A
Handout Number 36
RUNGE-KUTTA METHODS
(Section 25.3, page 701 in the 5
th
ed. or page 727 in the 6
th
ed.)
Advantage of Taylor methods of order
n
•
global truncation error of
)
(
n
h
O
insures high accuracy (even for
n
= 3, 4 or 5)
Disadvantage
•
high order derivatives of
))
(
,
(
x
y
x
f
may be difficult and expensive to evaluate.
Runge-Kutta methods
are higher order formulas (they can have any order
≥
1
) that
require function evaluations only of
))
(
,
(
x
y
x
f
, and not
of any of its derivatives.
This is accomplished using the Taylor polynomial for a function of 2 variables:
)
,
(
)
,
(
)
,
(
)
,
(
y
x
f
k
y
x
f
h
y
x
f
k
y
h
x
f
y
x
+
+
=
+
+
L
+
+
+
+
+
+
+
+
)
,
(
6
)
,
(
2
)
,
(
2
)
,
(
6
)
,
(
2
)
,
(
)
,
(
2
3
2
2
3
2
2
y
x
f
k
y
x
f
hk
y
x
f
k
h
y
x
f
h
y
x
f
k
y
x
f
hk
y
x
f
h
yyy
xyy
xxy
xxx
yy
xy
xx
where
y
x
f
f
x
f
f
xy
x
∂
∂
∂
≡
∂
∂
≡
2
,
,
etc.
The derivation
of Runge-Kutta methods and an understanding of why they work
requires
the Taylor polynomial for a function of 2 variables, but this Taylor polynomial is not
required to use these methods to numerically approximate the solution of a differential
equation.
A second order Runge-Kutta formula is derived in the textbook on page 703; however,
we will not consider their derivation, only their form and how to use them.
Runge-Kutta methods are so-called
one-step methods
(as also are Euler’s method and all
Taylor methods): that is, they are of the form (see (25.28) on p. 701 in 5
th
ed. or p. 727 in
6
th
ed.)
)
,
,
(
1
h
y
x
h
y
y
i
i
i
i
Φ
+
=
+