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Computational Techniques in Theoretical Physics Homework 12

Apply the Verlet algorithm to a one-dimensional simple harmonic oscillator of unit mass and spring constant, Newton's equation d²x/dt² = - x becomes:

x_n+1 - 2x_n + x_n-1 = - h² x_n

where h is step size, x_n = x(t=nh) is the value at time nh. Solve the difference equation exactly and compare its solution with that of the differential equation. Hint: look for solution of the form x_n=Re{A exp(i nh w)}.
Due to discretization, the energy is not exactly conserved. Can you construct an exactly conserved quantity which differs from total energy only by higher orders in h?

Give a detailed proof of the second order and forth order Runge-Kuttta method.