A Randomized Runge-Kutta Method for time-irregular delay differential equations

TL;DR

This paper introduces a randomized two-stage Runge-Kutta method for delay differential equations with irregular time dependence, analyzing its convergence and demonstrating its effectiveness through numerical experiments.

Contribution

It develops a novel randomized Runge-Kutta scheme tailored for DDEs with H"older continuous time components and provides theoretical error bounds.

Findings

The scheme achieves convergence in the $L^p$-norm.

Error bounds are established for the proposed method.

Numerical experiments confirm the method's effectiveness.

Abstract

In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions $f=f(t,x,z)$ that are Lipschitz continuous with respect to $x$ but only H\"older continuous with respect to $(t,z)$. We give a construction of the randomized two-stage Runge-Kutta scheme for DDEs and investigate its upper error bound in the $L^p(\Omega)$-norm for $p\in [2,+\infty)$. Finally, we report on results of numerical experiments.