Subject: addendum to v88.17

From: [email-masked] (The NA-NET)
Date: Sun, 24 Apr 88 11:03:32 PDT

>From [email-masked] Sun, 17 Apr 88 17:35:05 EDT
From: Nick Trefethen <[email-masked]>
Subject: Super-Accurate ODE Methods
To: [email-masked]

<This was overlooked for digest v88.17.  -mark>

I have a question about super-accurate numerical solution
of o.d.e.'s; can anyone advise?

Gerald Sussman, a professor in the M.I.T. Dept. of Electrical
Engineering and Computer Science, has built a special-purpose
machine for simulating the long-term evolution of the
outer planets of the solar system.   With this device he
and his colleagues have attempted to solve o.d.e.'s to perhaps
unprecedented accuracy: a few percent error over a time integration
of 200,000,000 years, which corresponds to around 20,000,000
orbits of Jupiter (the fastest time scale in the problem).

Here's a sketch of a typical calculation:

   Dimension of system of o.d.e.'s: 36
   Linear multistep formula: 12th-order Stoermer formula
   Step size: about 1/100th of the period of Jupiter
   Number of steps: O(10**9)   (!!)
   Final error: a few percent

My question is, can one do better than to use this 12th-order
Stoermer formula?  One hundred points per wavelength seems a
very large number to me, suggesting that a higher-order method
of some kind might be more efficient.  But Sussman claims that
in their experience, higher-order Stoermer formulas introduce
such large rounding errors, due to the large oscillatory
coefficients, that the final accuracy is reduced.

Are there other methods for super-accurate o.d.e. calculations
that ought to be considered?

                                  Nick Trefethen

   mail:  L. N. Trefethen, Dept. of Math., M.I.T., Cambridge, MA 02140
 e-mail:  [email-masked]  OR  [email-masked]
  phone:  617-253-4986