April 11 2008

MAIN POINTS
The text defines truncation error, a measure between the approximation and the exact solution of the differential equation. This error is shown to have behavior O(h) for Euler's method, where h is the value at which the function is computed.
The text then extends Euler's method. Euler's method is based on the n=1 Taylor polynomial, and the text extends this method to any value of n. This creates the "Taylor method of order n," which is just a substitution of the nth Taylor polynomial (sans error term) into Euler's method. This involves calculating n-1 derivatives, which the book is nice enough to carry out for 2nd and 4th order Taylor methods. The error of the 4th order method is impressively low.

CHALLENGES
How is this truncation error different from errors we've studied before? How can it measure the difference between the exact solution and the approximation if we don't know the exact solution? What is the significance of the final theorem— why does error seem to grow larger as the order of the method increases by (O(h^n))?

REFLECTION
Do you ever use this method when you're solving differential equations with a computer, or do you use the Runge-Kutta method from the next chapter? I'd like to see some real uses of differential equations, because I've only seen somewhat trivial ones. Are there any demos?

If you're doing anything like this with your lab over the summer, I'd like to check it out some time— I'll be staying here doing an internship (in scientific comp!) at the U.