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arXiv · 2512.01601

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

Wenbin Chen·Zhaohui Fu·Shun Wang·Xiaoming Wang
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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.

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Wenbin Chen, Zhaohui Fu, Shun Wang, Xiaoming Wang. 2025-12-01. Convergence of long-time stable variable-step arbitrary order ETD-MS scheme for gradient flows with Lipschitz nonlinearity. https://arxiv.org/abs/2512.01601

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