Abstract: A family of eighth order and sixth order P-stable methods for solving second order initial value problems is considered. The nonlinear algebraic system, which results on applying one of the methods in this family to a nonlinear differential system, may be solved by using a modified Newton method. The present study, introduces a local error estimation technique based on the derivation of suitable formula pairs. Thus, to obtain the local error estimate, we compute two approximations of the solution, one with a sixth order method and the other with an eighth order method. The error estimate is then obtained by subtracting our two approximations. The methods in each pair are chosen to have certain features in common. They have the same iteration matrix and some of the function evaluations are common to both methods. Finally numerical results are presented to illustrate our local error estimation technique.