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 $N$.