Euler scheme for approximation of solution of nonlinear ODEs under inexact information

TL;DR

This paper analyzes the Euler scheme's error when approximating solutions to nonlinear ODEs under nonstandard conditions and noisy data, providing optimality results and numerical experiments.

Contribution

It introduces an analysis of the Euler scheme's error under nonstandard assumptions and noisy information, including optimality and numerical validation.

Findings

Error bounds for Euler scheme under nonstandard conditions

Optimality results for approximation accuracy

Numerical experiments confirming theoretical analysis

Abstract

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local H\"older continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.