  A time-stepping scheme with adaptivity in both the step size and the
integration order is presented in the context of non-equilibrium dynamics
described via Kadanoff-Baym equations. The accuracy and effectiveness of the
algorithm are analysed by obtaining numerical solutions of exactly solvable
models. We find a significant reduction in the number of time-steps compared to
fixed-step methods. Due to the at least quadratic scaling of Kadanoff-Baym
equations, reducing the amount of steps can dramatically increase the
accessible integration time, opening the door for the study of long-time
dynamics in interacting systems. A selection of illustrative examples is
provided, among them interacting and open quantum systems as well as classical
stochastic processes. An open-source implementation of our algorithm in the
scientific-computing language Julia is made available.

Strongly Correlated Electrons;Quantum Gases;Computational Physics