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Assuming that you are integrating a set
of ordinary differential equations: first of all,
your friends problem appears to be one of "stability".
he may be using a very large increment in his
solutions. If he uses small enough increments, all
the integrators ought to give the same answers, unless
there is a mistake in the coding.
Seondly, some integrators are not suitable for certain
problems. Milne's method, for instance has a 
"parasitic instability" for first order odes.
Almost all integrators in existence will cause problems
if the system of equations has one exponentially growing
solution.

A good basic reference for ODEs is
Gerald and Wheatley: Applied Numerical Analysis.
(I dont know the publisher and location).

A slightly more advanced text is
Lambert J.D (1973) "Computational Methods in
Ordinary Differential Equations", Wiley, London
and New York.

Lambert has recently published a book on
Computational Methods for Systems of Equations.
(1991). I do not know the publisher and location.


As regards the truncation errors associated with various
methods, here is a brief overview.


Explicit Methods

1) Forward Euler   : First order accurate
2) Runge-Kutta     : First - Sixth order accurate
                     (depends on which particular RK is used)
3)Adams-Moulton    : Fourth order accurate


Implicit Methods  
1) Backward Euker  : First order accurate
2) Crank-Nicolson  : Second order accurate

Hope that helps.

Cheers

Makarand

In article <1hppagINNb5f@rave.larc.nasa.gov> makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes:
>
>Lambert has recently published a book on
>Computational Methods for Systems of Equations.
>(1991). I do not know the publisher and location.
>

Lambert, J.D.,
_Numerical_Methods_for_Ordinary_Differential_Systems_ (The Initial
Value Problem), John Wiley & Sons Ltd., Baffins Lane, Chichester, West
Sussex PO19 1UD, England, 1991.

>
>As regards the truncation errors associated with various
>methods, here is a brief overview.
>
>
>Explicit Methods
>
>1) Forward Euler   : First order accurate
>2) Runge-Kutta     : First - Sixth order accurate
>                     (depends on which particular RK is used)

Runge-Kutta methods of (much) higher orders have also been derived.

>3)Adams-Moulton    : Fourth order accurate

Adams-Moulton formulas exist for all orders.

It is usually best to try to use a polished general-purpose integrator whenever
possible.  It can be replaced with something tailored to the problem later, 
if necessary.  There are some good Runge-Kutta and linear multistep codes on
Netlib (send email to netlib@ornl.gov).
-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)