New Runge-Kutta algorithms are developed which determine the solution of a system of ordinary differential equations at any point within a given integration step, as well as at the end of each step.

Laurent Jay, Symplectic Partitioned Runge-Kutta Methods for Constrained Hamiltonian Systems, SIAM Journal on Numerical Analysis, Vol. 33, No. 1 (Feb., 1996), pp. 368-387 ...