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

136 CHAPTER 10. ORDINARY DIFFERENTIAL EQUATIONS
For example, Euler’s Method, equation 10.3 can be written as
X(t + h) ← X(t) + hF (t, X(t)) .
As in 1D, this method is derived from approximating X by its linearization.
In fact, our general strategy of using Taylor’s Theorem also carries through without change.
That is we can use the k th order method to step as follows:
X(t + h) ← X(t) + hX ′ (t) + h2
2 X′′ (t) + . . . + hk
k! X(k) (t).
Additionally we can write the Runge-Kutta Methods as follows:
Order two:
K 1 ← hF (t, X)
K 2 ← hF (t + h, X + K 1 )
X(t + h) ← X(t) + 1 2 (K 1 + K 2 ) .
Order four:
K 1 ← hF (t, X)
(
K 2 ← hF t + 1 2 h, X + 1 )
2 K 1
(
K 3 ← hF t + 1 2 h, X + 1 )
2 K 2
K 4 ← hF (t + h, X + K 3 )
X(t + h) ← x(t) + 1 6 (K 1 + 2K 2 + 2K 3 + K 4 ) .
Example Problem 10.8. Consider the system of ODEs:
⎧
⎪⎨
⎪⎩
x ′ 1 (t) = x 2 − x 2 3
x ′ 2 (t) = t + x 1 + x 3
x ′ 3 (t) = x 2 − x 2 1
x 1 (0) = 1,
x 2 (0) = 0,
x 3 (0) = 1.
Approximate X(0.1) by taking a single step of size 0.1, for Euler’s Method, and the Runge-Kutta
Method of order 2.
Solution: We write
⎡
x 2 (t) − x 2 3 (t) ⎤ ⎡ ⎤
1
F (t, X(t)) = ⎣ t + x 1 (t) + x 3 (t) ⎦ X(0) = ⎣ 0 ⎦
x 2 (t) − x 2 1 (t)
1
Thus we have
⎡
F (0, X(0)) = ⎣
−1
2
−1
⎤
⎦