TY - JOUR T1 - Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation AU - Cheng , Qing AU - Wang , Cheng JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1318 EP - 1354 PY - 2021 DA - 2021/08 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0297 UR - https://global-sci.org/intro/article_detail/aamm/19425.html KW - Epitaxial thin film equation, Fourier pseudo-spectral approximation, the scalar auxiliary variable (SAV) method, Crank-Nicolson temporal discretization, energy stability, optimal rate convergence analysis. AB -
A second order accurate (in time) numerical scheme is analyzed for the slope-selection (SS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. To make the numerical scheme linear while preserving the nonlinear energy stability, we make use of the scalar auxiliary variable (SAV) approach, in which a modified Crank-Nicolson is applied for the surface diffusion part. The energy stability could be derived a modified form, in comparison with the standard Crank-Nicolson approximation to the surface diffusion term. Such an energy stability leads to an $H^2$ bound for the numerical solution. In addition, this $H^2$ bound is not sufficient for the optimal rate convergence analysis, and we establish a uniform-in-time $H^3$ bound for the numerical solution, based on the higher order Sobolev norm estimate, combined with repeated applications of discrete Hölder inequality and nonlinear embeddings in the Fourier pseudo-spectral space. This discrete $H^3$ bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method. A few numerical experiments are also presented, which confirm the efficiency and accuracy of the proposed scheme.