Nonsmooth data error estimates for exponential Runge-Kutta methods and applications to split exponential integrators

TL;DR

This paper develops error estimates for exponential Runge-Kutta methods applied to parabolic equations with nonsmooth initial data, extending the analysis to abstract semilinear evolution equations and specific PDEs like Allen-Cahn and Burgers' equation, with applications to split exponential integrators.

Contribution

It introduces new nonsmooth data error bounds for exponential Runge-Kutta methods in a general abstract framework, including practical PDE applications.

Findings

Error bounds for nonsmooth initial data

Convergence results for split exponential integrators

Application to Allen-Cahn and Burgers' equations

Abstract

We derive error bounds for exponential Runge-Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in a framework of abstract semilinear evolution equations with operators having non-dense domain. In particular, we investigate nonsmooth data error estimates for the Allen-Cahn and the Burgers' equation. As an application, we apply these nonsmooth data error estimates to split exponential integrators and derive a convergence result in terms of the data.