Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection

Abstract

In this work, we are concerned with the stability and convergence analysis of the second order BDF (BDF2) scheme with variable steps for the molecular beam epitaxial model without slope selection. We first show that the variable-step BDF2 scheme is convex and uniquely solvable under a weak time-step constraint. Then we show that it preserves an energy dissipation law if the adjacent time-step ratios $r_k:={\tau }_k/{\tau }_{k-1}<\mathrm{3.561.}$ Moreover, with a novel discrete orthogonal convolution kernels argument and some new discrete convolutional inequalities, the $L^2$ norm stability and rigorous error estimates are established, under the same step-ratios constraint that ensuring the energy stability., i.e., $0<r_k<\mathrm{3.561.}$ This is known to be the best result in literature. We finally adopt an adaptive time-stepping strategy to accelerate the computations of the steady state solution and confirm our theoretical findings by numerical examples.

arXiv comment20 pages, 14 figures, 1 table

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Cited by in corpus (5)

• Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model
• Stability of variable-step BDF2 and BDF3 methods
• $L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model
• An efficient two-grid fourth-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation
• A new symmetric linearly implicit exponential integrator preserving polynomial invariants or Lyapunov functions for conservative or dissipative systems