## Extremely useful for second-order DEs in which

*d*2x/*dt*2 =*f*(*x*), e.g. SHM,*N*-body, etc.## Suppose

*x*is position, so*d*2*x*/*dt*2 is acceleration.## Procedure: define

*v*=*d*x/*dt*at the*midpoints*of the steps, i.e. velocities staggered wrt positions.- Define
*vi*+1/2 =*v*(*t*+ ½ *t*),*i*= 0, 1, 2, ... - Then advance
*xi*to*xi*+1 and*vi*+1/2 to*vi*+3/2:*xi*+1 =*xi*+*vi*+1/2 t*vi*+3/2 =*vi*+1/2 +*f*(*xi*+1) *t*

- Define

Also, now using t in place of