Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula id="j_cmam-2016-0048_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b1Aa"><jats:alternatives><jats:tex-math>{L^{p}(\Omega;{\mathbb{R}}^{d})}</jats:tex-math></jats:alternatives></jats:inline-formula>-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.</jats:p>