  This work is devoted to the obtaining of a new numerical scheme based in
quadrature formulas for the Lebesgue-Stieltjes integral for the approximation
of Stieltjes ordinary differential equations. This novel method allows us to
numerically approximate models based in Stieltjes ordinary differential
equations for which no explicit solution is known. We prove several theoretical
results related to the consistency, convergence and stability of the numerical
method. We also obtain the explicit solution of the Stieltjes linear ordinary
differential equation, and we use it to validate the numerical method. Finally,
we present some numerical results that we have obtained for a realistic
population model based in a Stieltjes differential equation.
