Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity

TL;DR

This paper proves the unconditional stability and optimal convergence of a variable-step high-order exponential time differencing scheme for gradient flows with Lipschitz nonlinearities, validated by numerical experiments.

Contribution

It introduces a unconditionally stable, high-order variable-step ETD-MS scheme with proven convergence for Lipschitz gradient flows, extending previous methods.

Findings

Unconditional energy stability of the scheme.

Optimal-order convergence under mild conditions.

Numerical validation on thin film epitaxial growth model.

Abstract

We analyze a variable-step extension of a family of arbitrarily high-order exponential time differencing multistep (ETD-MS) schemes recently developed by the authors. We prove that the schemes are unconditionally stable in the sense that a modified energy-representing a slight perturbation of the original energy-decreases monotonically over time, provided the nonlinearity is Lipschitz continuous in some appropriate sense. Moreover, we establish optimal-order convergence under mild conditions on the time-step size and local time-step ratio. Numerical experiments on the thin film epitaxial growth model without slope selection, employing a novel variable-step second-order scheme, validate the theoretical findings as well as its potential in developing highly efficient time-adaptive solution.