On simulations of the classical harmonic oscillator equation by   difference equations
  We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points (a lattice). Such exact discretization is found for an arbitrary lattice spacing. 
Jan L. Cieslinski, Boguslaw Ratkiewicz