Abstract:
We have compared several orders of Dormand El-Mikkawy and Prince [4] pairs applied to explicit Runge-Kutta Nystrom (ERKN) methods, hereafter clled families of Runge Kutta-Nystrom pairs comprising of the following 4 orders: ERKN4(3)4,ERKN(6(4)6,ERKN8(6)9 and ERKN12(10)17. In our comparison we realized that the pair ERKN 12(10)17 FM is the most efficient one of the family. We have shown these results in tables and graphs in section 3 where our main problem ( 2 body problem or Kepler’s problem ) is discussed in details, we then extended the Idea to many-body problem by coupling 8, 40 and even 1400 of Kepler’s problem and in particular we done the integration of many-body problems over a long term (period) of up to 1000. We used periodic property of the problem to compute the error at the end of each step, here called as End-point error, and kept an eye on cpu-time, and number of function evaluations along the way. All our programs are written in C language. In section 4 we would explain more about the measures we have taken in our discussion.