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with
≡
α
2
∂
∂
r
+
β
21
f
i
∂
D
2
∂y
.
(3.277)
Comparison with the Taylor expansion (3.272) shows that w
1
+
w
2
=
1, 2w
2
α
2
=
1,
2w
2
β
21
cient to determine the four parameters
uniquely. If we weight the two slopes equally, w
1
=
1. These three equations are insu
=
w
2
=
1/2andα
2
=
1, β
21
=
1.
Substitution in (3.274) then yields
2
hf
(
r
i
,y
i
)
hf
i
)
O
h
3
1
y
i
+
1
−
y
i
=
+
hf
(
r
i
+
h
,y
i
+
+
O
h
3
,
1
2
(
k
1
=
+
k
2
)
+
(3.278)
for
k
1
k
1
). This second-order accurate approxima-
tion has been obtained without calculation of derivatives of
f
(
r
,y). Such schemes of
integration, avoiding the calculation of derivatives, were first developed by Runge
(1895), with further contributions by Kutta (1901). Instead, two evaluations of
f
(
r
,y) are required for second-order accuracy. Higher-order accuracy can be
achieved by increasing the number of evaluations. Because of the underdetermined
nature of the parameters, a wide variety of integration formulae have been derived,
generally referred to as
Runge-Kutta methods
. The scheme (3.278) is one of several
second-order Runge-Kutta formulae.
The foregoing analysis can be generalised. For example, for fourth-order accur-
acy we write
=
hf
(
r
i
,y
i
),
k
2
=
hf
(
r
i
+
h
,y
i
+
y
i
+
1
−
y
i
=
w
1
k
1
+
w
2
k
2
+
w
3
k
3
+
w
4
k
4
,
(3.279)
with the four required evaluations,
k
1
=
hf
(
r
i
,y
i
),
k
2
=
hf
(
r
i
+
α
2
h
,y
i
+
β
21
k
1
),
(3.280)
k
3
=
hf
(
r
i
+
α
3
h
,y
i
+
β
31
k
1
+
β
32
k
2
),
k
4
=
hf
(
r
1
+
α
4
h
,y
i
+
β
41
k
1
+
β
42
k
2
+
β
43
k
3
).
The requisite Taylor expansions of
f
(
r
,y) in two variables are listed by Ralston and
Rabinowitz (1978, Sec. 5.8). Once again, comparison with the expansion (3.272)
gives an underdetermined system of equations in the unknown parameters, allow-
ing a wide variety of fourth-order accurate schemes.
Instead, we concentrate on the most commonly used fourth-order Runge-Kutta
scheme,
1
6
(
k
1
y
i
+
1
−
y
i
=
+
2
k
2
+
2
k
3
+
k
4
),
(3.281)
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