Randomized Runge-Kutta method -- stability and convergence under inexact information

TL;DR

This paper introduces a randomized two-stage Runge-Kutta method for solving ODEs with inexact information, demonstrating its optimality and analyzing its stability and convergence properties through theoretical and numerical studies.

Contribution

The paper establishes the optimality of a randomized Runge-Kutta scheme for ODE approximation under noisy data and thoroughly investigates its stability regions.

Findings

The randomized scheme is proven to be optimal among algorithms with noisy information.

Numerical experiments confirm the theoretical stability and convergence results.

The stability regions of the optimal method are rigorously characterized.

Abstract

We deal with optimal approximation of solutions of ODEs under local Lipschitz condition and inexact discrete information about the right-hand side functions. We show that the randomized two-stage Runge-Kutta scheme is the optimal method among all randomized algorithms based on standard noisy information. We perform numerical experiments that confirm our theoretical findings. Moreover, for the optimal algorithm we rigorously investigate properties of regions of absolute stability.