Re: Euler method vindicated

carlos_at_colorado.edu
Date: 12/11/04 

Date: 10 Dec 2004 22:31:30 -0800

khufu wrote:
> Thanks. Is it "Solving Ordinary Differential Equations" volumes 1 and
2
> that you recommend?

Yes. Both volumes are expensive though.

Here is the method that C. W. Gear calls in his 1971 book
"midpoint method" to integrate y'=f(y,t) (h=stepsize)

y_(n+1/2) = y_n + (h/2) f(y_n,t_n)
y_(n+1) = y_n + h f(y_(n+1/2),t+h/2) (*)

He calls the variant

yP = y_n + h f(y_n,t_n)
y_(n+1) = y_n + (h/2) [f(y_n,t_n)+f(yP,t+h)]

the "modified trapezoid method". Both are 2nd order and explicit. To
compound the confusion here are other names: Euler-Cauchy, Improved
polygon, Improved Euler, Modified Euler, Heun, Midpoint rule, 2nd-order
Runge-Kutta. Some authors switch names.

I used (*) in my 1966 thesis to do computational plasticity, and
remember lifting it from Henrici's 1964 book.

The time-reversible, symplectic variants receive additional names:
Stormer, Verlet, Shake, Rattle, Lobatto pair, ... See H&W vol II,
VII.8.