QMC methods for the solution of delay differential equations

In this paper the quasi-Monte Carlo methods for Runge–Kutta solution techniques of differential equations, which were developed by Stengle, Lécot, Coulibaly and Koudiraty, are extended to delay differential equations of the form y′(t)=f(t,y(t),y(t−τ(t))). The retarded argument is approximated by interpolation, after which the conventional (quasi-)Monte Carlo Runge–Kutta methods can be applied. We give a proof of the convergence of this method and its order in a general form, which does not depend on a specific quasi-Monte Carlo Runge–Kutta method. Finally, a numerical investigation shows that similar to ordinary differential equations, this quasi-randomized method leads to an improvement for heavily oscillating delay differential equations, compared even to high-order Runge–Kutta schemes.