Quasi-randomized numerical methods for systems with coefficients of bounded variation

We propose a family of numerical schemes to solve the initial value problem for a system of differential equations y′(t)=f(t,y(t)) in which f is smooth in space (y), but only of bounded variation in time (t). The family is akin to the Runge–Kutta family. However, the discretization with respect to time only retains mean properties. The means are estimated with Monte Carlo simulation. We analyze first and second order methods which use quasi-random point sets for the simulation. Error bounds are derived which involve powers of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. The numerical results indicate that by using quasi-random points in place of pseudo-random points we are able to obtain smaller errors.