Discretization by euler's method for regular lagrangian flow

TL;DR

This paper analyzes the explicit Euler scheme's convergence to regular Lagrangian flows for ODEs with non-Lipschitz vector fields, establishing a convergence order of 1/2.

Contribution

It proves the convergence of the Euler scheme to Diperna-Lions flows for non-Lipschitz vector fields, with a convergence order of 1/2.

Findings

Proves convergence of Euler scheme to regular Lagrangian flows.

Establishes the convergence order as 1/2.

Addresses numerical analysis for non-Lipschitz ODEs.

Abstract

This paper is concerned with the numerical analysis of the explicit Euler scheme for ordinary differential equations with non-Lipschitz vector fields. We prove the convergence of the Euler scheme to regular lagrangian flow (Diperna-Lions flows) which is the right concept of the solution in this context. Moreover, we show that order of convergence is $\frac{1}{2}$.