A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution

We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge-Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method.