STABILITY AND CONVERGENCE OF A VARIABLE-STEP STABILIZED BDF2 STEPPING FOR THE MBE MODEL WITH SLOPE SELECTION

A second order variable-step stabilized convex splitting BDF2 time-stepping is investigated for the molecular beam epitaxy model with slope selection. The common Douglas-Dupont regularization term A tau(n)triangle(phi(n) - phi(n-1)) with a properly stabilized parameter A> 0 is considered such that the numerical scheme preserves the discrete energy dissipation law unconditionally. The present error analysis is essentially different from the traditional energy method [W. Chen, X. Wang, Y. Yan, and Z. Zhang, SIAM J. Numer. Anal., 57:495-525, 2019] by combining the L-2 norm with H(1 )norm analysis, which always require relatively stringent step-ratio restriction. Our main tools are the so called discrete orthogonal convolution kernels and the associated convolution embedding inequalities. Under the adjacent step ratios constraint tau(k)/tau(k-1) <4.864, which is stemmed from the positive definiteness of BDF2 convolution kernels, an optimal L-2 norm error estimate is achieved for the first time by carefully handling the Douglas-Dupont regularization term. Numerical experiments are presented to support our theoretical results.