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

140 CHAPTER 10. ORDINARY DIFFERENTIAL EQUATIONS
for any r ≥ 0, and t 0 ∈ (0, 2π] . Given an initial value, the parameters r and t 0 are uniquely
determined. Thus the trajectory of X over time is that of an ellipse in the plane. 2 Thus the
family of such ellipses, taking all r ≥ 0, form the phase curves of this ODE. We would expect the
approximation of an ODE to never cross a phase curve. This is because the phase curve represents
the
Euler’s Method and the second order Runge-Kutta Method were used to approximate the ODE
with initial value
[ ] 1.5
X (0) = .
1.5
Euler’s Method was used for step size h = 0.005 for 3000 steps. The Runge-Kutta Method was
employed with step size h = 0.015 for 3000 steps. The approximations are shown in Figure 10.7.
Given that the actual solution of this initial value problem is an ellipse, we see that Euler’s
Method performed rather poorly, spiralling out from the ellipse. This must be the case for any step
size, since Euler’s Method steps in the direction tangent to the actual solution; thus every step of
Euler’s Method puts the approximation on an ellipse of larger radius. Smaller stepsize minimizes
this effect, but at greater computational cost.
The Runge-Kutta Method performs much better, and for larger step size. At the given resolution,
no spiralling is observable. Thus the Runge-Kutta Method outperforms Euler’s Method, and
at lesser computational cost. 3
2 In the case r = 0, the ellipse is the “trivial” ellipse which consists of a point.
3 Because the Runge-Kutta Method of order two requires two evaluations of F , whereas Euler’s Method requires
only one, per step, it is expected that they would be ‘evenly matched’ when Runge-Kutta Method uses a step size
twice that of Euler’s Method.