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

TL;DR

This paper investigates the convergence and stability properties of a randomized implicit two-stage Runge-Kutta scheme, demonstrating its asymptotic and probabilistic A-stability but not mean-square stability.

Contribution

It introduces a randomized version of the implicit RK2 scheme and analyzes its stability in various probabilistic senses, extending classical stability concepts.

Findings

The randomized scheme improves convergence rate over deterministic schemes.

It is asymptotically and in probability A-stable.

The scheme is not mean-square A-stable.

Abstract

We randomize the implicit two-stage Runge-Kutta scheme in order to improve the rate of convergence (with respect to a deterministic scheme) and stability of the approximate solution (with respect to the solution generated by the explicit scheme). For stability analysis, we use Dahlquist's concept of A-stability, adopted to randomized schemes by considering three notions of stability: asymptotic, mean-square, and in probability. The randomized implicit RK2 scheme proves to be A-stable asymptotically and in probability but not in the mean-square sense.