numerical solution of ODEs

From: Igor (ikuchmienko_at_mail.ru)
Date: 11/20/04 

Date: Sat, 20 Nov 2004 13:55:15 +0000 (UTC)

I am trying to get familiar with basic strategies of consructing
methods for numerical solution of Ordinary Differential Equations,
and the state of the art in this field.

I have read two volumes of 'Solution of Ordinary Differential
Equations' by Hairer, Wanner so far. As I've understood, the basic
strategy proposed there is in brief as follows. We choose some form
of a method where coefficients are unknown (one of the most general
forms is general linear methods, which include Runge-Kutta and
multistep methods as particular cases). Then we are trying to find
coefficients from satisfying a set of formal conditions (most popular
are order conditions and various sorts of stabilities, like D,A or L-
stability).

Is there a radically different approach? Could you recommend any
books on it? As an example, I've heard that Lie groups can be somehow
applied numerically.

I've also read a book by Hackbusch on numerical solution of integral
equations. Since an ODE has an equivalent formulation as a Volterra
integral equation of the second kind, a natural question is how
successful can applying methods for solution of integral equations to
differential equations be? Some of them are analogous, for example, a
collocation method which exists for both integral and differential
equations (and which is also an implicit Runge-Kutta method with an
arbitrarily high order). Its drawback is that it leads to large dense
matrices, but in integral equations it's facilitated by use of
special solvers like a multi-grid method or representation in wavelet
basis. Why not applying these techniques for ODEs? If they ARE
applied, where can I read about it?

Thanks.