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

Abstract

Abstract. This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with timeirregular 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 L p (Ω; 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.

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

“…Next, we collect some well-known error estimates for the approximation of the semigroup (S(t)) t∈[0,T ] ⊂ L(H). Recall the definition of S k,h from (20). For a proof of the first two error bounds in Lemma 5.2 we refer to [32,Chapter 7].…”

“…Proof. First, we replace S n−j+1 k,h by its piecewise constant interpolation S k,h defined in (20). After adding and subtracting a few additional terms we arrive at n j=1 tj tj−1…”

“…Lemma 5.9 is applicable to the first term on the right hand side and it remains to give a bound for the second term. To this end, we recall the operators S k,h andĝ defined in (20) and (44), respectively. Inserting these operators then yields max n∈{1,...,N k } n j=1 S n−j+1 k,h…”

On a Randomized Backward Euler Method for Nonlinear Evolution Equations with Time-Irregular Coefficients

“…The time-irregular case studied in the present paper was first investigated in [38,39]. See also [26,30] for a more recent exposition of explicit randomized schemes.…”

“…In order to deal with possibly stiff ODEs we consider a randomized version of the backward Euler method and prove its well-posedness and stability under a one-sided Lipschitz condition. In addition, we require only local Lipschitz conditions with respect to the state variable in order to obtain estimates on the local truncation error, thereby extending results from [30]. We also avoid any (local) boundedness condition on f as, for example, in [12,26].…”