An algorithm which implements these equations is included below.

• Choose the initial position and speed, with the most common choice \( v(t=0)=0 \) and some fixed value for the position.
• Choose the method you wish to employ in solving the problem.
• Subdivide the time interval \( [t_i,t_f] \) into a grid with step size

$$ h=\frac{t_f-t_i}{N}, $$ where \( N \) is the number of mesh points.

• Calculate now the total energy given by

$$ E_0=\frac{1}{2}kx(t=0)^2=\frac{1}{2}k. $$

• The Runge-Kutta method is used to obtain \( x_{i+1} \) and \( v_{i+1} \) starting from the previous values \( x_i \) and \( v_i \).
• When we have computed \( x(v)_{i+1} \) we upgrade \( t_{i+1}=t_i+h \).
• This iterative process continues till we reach the maximum time \( t_f \).
• The results are checked against the exact solution. Furthermore, one has to check the stability of the numerical solution against the chosen number of mesh points \( N \).