🚀 go-pugleaf

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