# Error analysis of randomized Runge-Kutta methods for differential equations with time-irregular coefficients

## TL;DR

This paper provides an error analysis and convergence rates for two randomized Runge-Kutta methods applied to ODEs with irregular time-dependent coefficients, including those with singularities or weak regularity.

## Contribution

It introduces precise error bounds and convergence rates for randomized Runge-Kutta schemes handling time-irregular coefficients in ODEs, extending applicability to Carathéodory and singular cases.

## Key findings

- Derived $L^p$-norm error bounds for the methods.
- Established almost sure convergence rates.
- Validated results through numerical experiments.

## Abstract

This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carath\'eodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the $L^p(\Omega;\mathbb{R}^d)$-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.