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 Carathéodory 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 -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.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: FOR 2402
Funding statement: This research was carried out in the framework of Matheon supported by Einstein Foundation Berlin. The authors also gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft through the research unit FOR 2402 (Rough paths, stochastic partial differential equations and related topics) at TU Berlin.
The authors like to thank Wolf-Jürgen Beyn, Monika Eisenmann, Mihály Kovács, and Stig Larsson for inspiring discussions and helpful comments.
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