Convergence and stability of randomized implicit two-stage Runge–Kutta schemes

The randomized versions of implicit two-stage Runge–Kutta are considered to improve the rate of convergence (compared to a deterministic scheme) and stability of the approximate solution (contrasted to the solution generated by the explicit scheme). Dahlquist’s concept of A-stability is adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme is A-stable asymptotically and in probability but not in the mean-square sense.