Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

10.2. RUNGE-KUTTA METHODS 133
We now examine the “proper” choices of the constants. Note that w 1 = 1, w 2 = 0 corresponds
to Euler’s Method, and does not require computation of K 2 . We will pick another choice.
Also notice that the definition of K 2 should be related to Taylor’s theorem in two dimensions.
Let’s look at it:
K 2 /h = f(t + αh, x + βK 1 )
= f(t + αh, x + βhf(t, x))
= f + αhf t + βhff x + 1 (
α 2 h 2 f tt (¯t, ¯x) + αβfh 2 f tx (¯t, ¯x) + β 2 h 2 f 2 f x x(¯t, ¯x) ) .
2
Now reconsider our step:
x(t + h) = x(t) + w 1 K 1 + w 2 K 2
= x(t) + w 1 hf + w 2 hf + w 2 αh 2 f t + w 2 βh 2 ff x + O ( h 3) .
= x(t) + (w 1 + w 2 ) hx ′ (t) + w 2 h 2 (αf t + βff x ) + O ( h 3) .
= x(t) + (w 1 + w 2 ) hx ′ (t) + w 2 h 2 (αf t + βff x ) + O ( h 3) .
If we happen to choose our constants such that
then we get
w 1 + w 2 = 1, αw 2 = 1 2 = βw 2,
x(t + h) = x(t) + hx ′ (t) + 1 2 h2 (f t + ff x ) + O ( h 3)
= x(t) + hx ′ (t) + 1 2 h2 x ′′ (t) + O ( h 3) ,
i.e., our choice of the constants makes the approximate x(t + h) good up to a O ( h 3) term, because
we end up with the Taylor’s series expansion up to that term. cool.
The usual choice of constants is α = β = 1, w 1 = w 2 = 1 2
. This gives the second order Runge-
Kutta Method :
This can be written (and evaluated) as
x(t + h) ← x(t) + h 2 f(t, x) + h f(t + h, x + hf(t, x)).
2
K 1 ← hf(t, x)
K 2 ← hf(t + h, x + K 1 )
x(t + h) ← x(t) + 1 2 (K 1 + K 2 ) .
(10.4)
Another choice is α = β = 2/3, w 1 = 1/4, w 2 = 3/4. This gives
x(t + h) ← x(t) + h 4 f(t, x) + 3h 4 f (
t + 2h 3 , x + 2h 3 f(t, x) )
.
The Runge-Kutta Method of order two has error term O ( h 3) . Sometimes this is not enough
and higher-order Runge-Kutta Methods are used. The next Runge-Kutta Method is the order four