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

Fabio V. Difonzo, Paweł Przybyłowicz, Yue Wu, Xinheng Xie

TL;DR

The paper tackles time-irregular delay differential equations where the right-hand side is Lipschitz in the state but Hölder in time and delay arguments. It introduces a randomized two-stage Runge-Kutta scheme (ARRK) tailored to Carathéodory-type DDEs, establishes existence/uniqueness of solutions, and proves an $L^p$-error bound of the form $\|\max_{0\le i\le N}|x(t_i^j)-y_i^j|\|_p \le C_{p,j} h^{\alpha^j\rho}$ with $\rho=\tfrac12+\min\{\gamma,\alpha\}$. The numerical experiments demonstrate that ARRK outperforms randomized Euler in accuracy for small step sizes, justifying the additional cost of the intermediate computations, and the results corroborate the theoretical convergence behavior across varied parameters. Overall, the work provides a rigorous, implementable framework for solving Carathéodory DDEs with time-irregular delays, expanding the toolbox of numerical methods for such systems.

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 Hölder 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 $L^p(Ω)$-norm for $p\in [2,+\infty)$. Finally, we report on results of numerical experiments.

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

TL;DR

The paper tackles time-irregular delay differential equations where the right-hand side is Lipschitz in the state but Hölder in time and delay arguments. It introduces a randomized two-stage Runge-Kutta scheme (ARRK) tailored to Carathéodory-type DDEs, establishes existence/uniqueness of solutions, and proves an -error bound of the form with . The numerical experiments demonstrate that ARRK outperforms randomized Euler in accuracy for small step sizes, justifying the additional cost of the intermediate computations, and the results corroborate the theoretical convergence behavior across varied parameters. Overall, the work provides a rigorous, implementable framework for solving Carathéodory DDEs with time-irregular delays, expanding the toolbox of numerical methods for such systems.

Abstract

In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions that are Lipschitz continuous with respect to but only Hölder continuous with respect to . We give a construction of the randomized two-stage Runge-Kutta scheme for DDEs and investigate its upper error bound in the -norm for . Finally, we report on results of numerical experiments.
Paper Structure (9 sections, 6 theorems, 59 equations, 3 figures, 1 table)

This paper contains 9 sections, 6 theorems, 59 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of solutions to Carathéodory DDEs
  4. Error of the randomized RK scheme
  5. Numerical experiments
  6. The implementation
  7. Example 1
  8. Example 2
  9. Appendix

Key Result

Theorem 3.1

Let $n\in\mathbb{N}\cup\{0\}$, $\tau\in (0,+\infty)$, $x_0\in\mathbb{R}^d$ and let $f$, $\varphi$ satisfy the assumptions ass:A1-ass:A4. Then there exists a unique absolutely continuous solution $x=x(x_0,f)$ to eq:DiscDDE2 such that for $j=0,1,\ldots,n$ we have where $K_{-1}:=\max_{t\in [-\tau,0]}|\varphi(t)|$ and then for all $j=0,1,\ldots,n$, $t,s\in [0,\tau]$ it holds

Figures (3)

  • Figure 1: Visual representation of Randomized RK method for different $j$.
  • Figure 2: The performance of solving DDE \ref{['eqn:DDEexample1']} via randomized Runge-Kutta and randomised Euler.
  • Figure 3: The performance of solving DDE \ref{['eqn:DDEexample2']} via randomized Runge-Kutta.

Theorems & Definitions (12)

  • Theorem 3.1
  • Remark 3.2
  • proof
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Lemma 6.1
  • Lemma 6.2
  • ...and 2 more