A Naive Variable Step Size Method
This demonstration illustrates the prinicpals behind
automatic step size adjustment
using a fairly crude, but easy to understand algorithm. It generates
an approximate solution to the initial value problem
Here is an outline of the algorithm. For the details see
It first does one step of Euler's method using a trial step size
h. Call the answer A1. It also does two steps of Euler method's with step
size h/2. Call the answer A2. The local truncation error introduced
in A1 is Kh**2 +O(h**3). The local truncation error introduced
in A2 is Kh**2/2 +O(h**3)=A1-A2+O(h**3). If the error per unit advance
in t, |A1-A2|/h, is smaller than the user specified error rate, the result
A2 is accepted and the procedure moves on to the next step. Otherwise
it repeats the current step with trial step size h'= .9*2*(error rate)/K.
- y'=exp( -8(t-.5)**2 )
- y(0)= exp(-2)
THIS DEMONSTRATION USES A LARGE WINDOW. SCROLL AND RESIZE YOUR
BROWSER WINDOW SO THAT YOU CAN SEE THE WHOLE WINDOW.
To execute one step click the "next" button. The upper window gives the
approximate solution constructed to date. The lower window gives a blown
up view of the current step.
To start over at t=0 click the "restart" button. The demonstration
provides a running commentary which at each step
tells you what computations were done.