An F-modulated stability framework for multistep methods
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We introduce a new 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 BDFk, 2≤ k ≤ 5 discretization schemes with explicit k^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 explicit maximal time step constraints which guarantee monotonic 𝐅-modulated energy dissipation.
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