Re: numerical solution of ODEs

From: John T Lowry (jlowry100_at_earthlink.net)
Date: 11/20/04 

Date: Sat, 20 Nov 2004 15:19:51 GMT

"Igor" <ikuchmienko@mail.ru> wrote in message
news:200411201219.iAKCJ4d26864@proapp.mathforum.org...
>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.

I suggest Lester Ford's book Differential Equations.

John Lowry
Flight Physics

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