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arXiv · 2111.05210

An F-modulated stability framework for multistep methods

Dong Li·Chaoyu Quan·Wen Yang
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Abstract

We introduce a new $\mathbf F$-modulated energy stability framework for general linear multistep methods. We showcase the theory for the two dimensional molecular beam epitaxy model with no slope selection which is a prototypical gradient flow with Lipschitz-bounded nonlinearity. We employ a class of representative BDF$k$, $2\le k \le 5$ discretization schemes with explicit $k^{\mathrm{th}}$-order extrapolation of the nonlinear term. We prove the uniform-in-time boundedness of high Sobolev norms of the numerical solution. The upper bound is unconditional, i.e. regardless of the size of the time step. We develop a new algebraic theory and calibrate nearly optimal and \emph{explicit} maximal time step constraints which guarantee monotonic $\mathbf F$-modulated energy dissipation.

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Dong Li, Chaoyu Quan, Wen Yang. 2021-11-09. An F-modulated stability framework for multistep methods. https://arxiv.org/abs/2111.05210

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