An F-modulated stability framework for multistep methods

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.

21 pages

References in corpus (3)

• Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials
• The energy technique for the six-step BDF method
• Effective maximum principles for spectral methods