Lie-Trotter method for abstract semilinear evolution equations

TL;DR

This paper provides a unified analysis of the Lie-Trotter method and related schemes for solving various semilinear evolution equations, establishing convergence results based on Lipschitz estimates in suitable Hilbert spaces.

Contribution

It introduces a comprehensive framework for analyzing Lie-Trotter and similar methods for semilinear problems, including convergence and linear convergence under regularity conditions.

Findings

Convergence of Lie-Trotter method in Hilbert spaces.

Extension to schemes like Strang and Ruth--Yoshida.

Linear convergence under additional regularity assumptions.

Abstract

In this paper we present a unified picture concerning Lie-Trotter method for solving a large class of semilinear problems: nonlinear Schr\"odinger, Schr\"oginger--Poisson, Gross--Pitaevskii, etc. This picture includes more general schemes such as Strang and Ruth--Yoshida. The convergence result is presented in suitable Hilbert spaces related with the time regularity of the solution and is based on Lipschitz estimates for the nonlinearity. In addition, with extra requirements both on the regularity of the initial datum and on the nonlinearity we show the linear convergence of the method.