An adaptive BDF2 implicit time-stepping method for the no-slope-selection epitaxial thin film model

In this paper we are concerned with the stability and convergence analysis of the second order backward differentiation formula (BDF2) scheme with variable time steps for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral method in physical domain. Under the adjoint time-step ratio condition r(k) := t(k)/t(k-1) < 4.864, the variable-step BDF2 scheme is uniquely solvable and stable in L-2-norm, also we establish a rigorous error estimate with a novel discrete orthogonal convolution kernels involved in the analysis. Finally, the adaptive time step strategy is used to accelerate the calculation of the steady-state solution, and the theoretical results are verified by numerical examples. In addition, the long time simulation results have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width.