Another set of popular algorithms, which are both numerically stable and easy to implement are the Verlet algorithms, with the velocity Verlet method as widely used in for example Molecular dynamics calculations. Consider again a second-order differential equation like Newton's second law, whose one-dimensional version reads $$ m\frac{d^2 x}{dt^2}= F(x,t), $$ which we rewrite in terms of two coupled differential equations $$ \frac{dx}{dt}=v(x,t) \hspace{1cm}\mathrm{and}\hspace{1cm} \frac{dv}{dt}=F(x,t)/m=a(x,t). $$