This paper presents a discretization scheme for Mayer's type optimal control problems of linear systems. The scheme is based on second order Volterra-Fliess approximations, and on an augmentation of the control variable in a control set of higher dimension. Compared with the existing results, it has the advantage of providing a higher order accuracy, which may make it more efficient when aiming for a certain precision. Error estimations (depending on the controllability index of the system at the solution) are proved by using a recent result about stability of the optimal solution with respect to disturbances. Numerical results are provided which show the sharpness of the error estimations.