Mathematica: Adams Bashforth-Moulton method and its errors

Kortirion

Greetings PF. I am new to the subject of numerical methods and I'm interested in using the Adams method in Mathematica, this one with a predictor-corrector algorithm, to numerically solve a system of differential equations (first-order system).

As I'm pretty green I was reading this page http://reference.wolfram.com/mathema...lvePlugIns.cdf and its section on the Adams method. I don't have the skill to make a more efficient algorithm than the one prescribed in there, so I just copy-pasted all the necessary code into my Mathematica notebook. With this I could use it within NDSolve by adding "Method -> AdamsBM".

It works, rolls ok with my system and when I take the difference in solutions of the "regular" unspecified method of NDSolve with this AdamsBM method, there's some difference depending on the working precision I tell AdamsBM to work in. So they really are different and this "working precision" plays some role.

What I'm really interested in at this point is the error of this numerical AdamsBM method. So in short - how do I calculate this error? How do I know that this AdamsBM is better than the other for example? I've read around a bit on the internet but couldn't find anything that fits my level of understanding regarding this subject.

Actually I have some other questions as well, but they are tied in with this question about the error of the method at hand. I think it's a good starting point.

Any advice or help is most appreciated, whether it's about the Adams method in general or any of its specifics. Thanks in advance!