Recently, Breitenbach and Borzì have proposed a sequential quadratic Hamiltonian method for solving elliptic optimal control problems in the optimize-then-discretize framework. They prove the monotonic convergence of the algorithm in the continuous space case. However, the properties of the discrete version of the iterative procedure have not been yet tackled with in the discretize-then-optimize framework. In this paper we present a center difference scheme for one-dimensional elliptic optimal control problems and prove that the scheme preserves the monotonic properties of the sequential quadratic Hamiltonian method. Numerical experiments show that the proposed algorithm is effective and convergent.