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

134 CHAPTER 10. ORDINARY DIFFERENTIAL EQUATIONS
method:
K 1 ← hf(t, x)
K 2 ← hf(t + h/2, x + K 1 /2)
K 3 ← hf(t + h/2, x + K 2 /2)
K 4 ← hf(t + h, x + K 3 )
x(t + h) ← x(t) + 1 6 (K 1 + 2K 2 + 2K 3 + K 4 ) .
(10.5)
This method has order O ( h 5) . (See http://mathworld.wolfram.com/Runge-KuttaMethod.html)
The Runge-Kutta Method can be extrapolated to even higher orders. However, the number
of function evaluations grows faster than the accuracy of the method. Thus the methods of order
higher than four are normally not used.
We already saw that w 1 = 1, w 2 = 0 corresponds to Euler’s Method. We consider now the case
w 1 = 0, w 2 = 1. The method becomes
x(t + h) ← x(t) + hf
(t + h 2 , x + h 2 f(t, x) )
.
This is called the modified Euler’s Method . Note this gives a different value than if Euler’s
Method was applied twice with step size h/2.
10.2.3 Examples
Example 10.6. Consider the ODE:
{
x ′ = (tx) 3 − ( x
x(1) = 1
Use h = 0.1 to compute x(1.1) using both Taylor’s Series Methods and Runge-Kutta methods
of order 2.
t
) 2
10.3 Systems of ODEs
Recall the regular ODE problem: find some x(t) such that
{ dx(t)
dt
= f (t, x(t)) ,
x(a) = c,
where f, a, c are given.
Sometimes the physical systems we are considering are more complex. For example, we might
be interested in the system of ODEs:
⎧
⎪⎨
⎪⎩
dx(t)
dt
= f (t, x(t), y(t)) ,
dy(t)
dt
= g (t, x(t), y(t)) ,
x(a) = c,
y(a) = d.