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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'(t) = f(t,y(t)). The function f is smooth in y and we suppose that f and D 1 y f are of bounded variation in t and that D 2 y 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.