Hi Ming, Presumably your ODE looks like y'' = f(y), given y(0) and y'(0). (I could be wrong.) Assuming your ODE does look like this, a quick way to use Kahan's (and my) composition schemes is to do as follows: 1) introduce x=y'. ODE becomes y' = x x' = f(y) given x(0) and y(0) 2) to advance from (t, x, y) to (t+h, X, Y), do Z = y +(h/2) x; X=x+h f(Z); Y = Z+(h/2) X; Call it [X,Y]=Q(h,x,y). This gives you a second order convergence method and it is sympletic. 3) Compose Q(h,x,y) to higher order sympletic integrators by repeatedly calling function Q(.,.,.). Composition constants can be found in the attached PDF file. High order schemes can be expensive. Best, -rencang