Quasi-randomized numerical methods for systems with coefficients of bounded variation (Q5938369)

This paper is concerned with the numerical solution of initial value problems for systems of first order differential equations \( y' = f (t,y)\) where \(f \) is smooth with respect to \( y \) but of bounded variation in \(t\). NEWLINENEWLINENEWLINEThe methods under consideration are formally similar to one- and two stage Runge-Kutta methods, however the discretization with respect to the variable \(t\) is carried out with Monte Carlo simulation. The author proposes first and second order methods that consider quasi random times for the simulation. NEWLINENEWLINENEWLINEIt is proved that, in spite of the random approximation in \(t\), error bounds in powers of the size of the step can be derived. Finally some numerical experiments with the linear non homogeneous test equation \( y' = y + \mu \sin ( \cos ( \lambda t))\) for several values of parameters \( \mu \) and \( \lambda \) are presented to show that a quasi random choice is preferred to the pseudo-random option in order to get smaller errors.