A quasi-randomized Runge-Kutta method

We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations $y^{\prime}(t) = f (t,y(t))$. The function $f$ is smooth in $y$ and we suppose that $f$ and $D_y^1f$ are of bounded variation in $t$ and that $D_{y}^2 f$ is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.