
Since x
i
is unknown after the initial step, it must be replaced by x
A
(i) in Equation 6.27 to yield the
difference equation for a numerical integrator. Using the definitions for k
1
and k
2
in Equations 6.14
and 6.15 and remembering that p ¼q ¼1 give
x
A
(i þ 1) ¼ x
A
(i) þ
T
2
{f [t
i
, x
A
(i)] þ f [t
i
þ T, x
A
(i) þ Tf [t
i
, x
A
(i)]]} (6:28)
Denoting x
A
(i) þTf [t
i
, x
A
(i)] by
^
x
A
(i þ 1) in Equation 6.28 gives
x
A
(i þ 1) ¼ x
A
(i) þ
T
2
{f [t
i
, x
A
(i)] þ f [t
i
þ T,
^
x
A
(i þ 1)]} (6:29)
You should recognize
^
x
A
(i þ 1) as the explicit Euler estimate of x
iþ1
in Equation 6.11 (see Figure
6.2). Hence, the explicit Euler (an RK-1 integrator) establishes the second point [t
i
þ T,
^
x
A
(i þ 1