Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation  

作　　者：Qing Cheng Cheng Wang 

机构地区：[1]Department of Mathematics,Purdue University,West Lafayette,IN 47907,USA [2]Department of Mathematics,University of Massachusetts,North Dartmouth,MA 02747,USA

出　　处：《Advances in Applied Mathematics and Mechanics》2021年第6期1318-1354,共37页应用数学与力学进展（英文）

基　　金：supported in part by NSF DMS-2012669(C.Wang).

摘　　要：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 H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 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.

关 键 词：Epitaxial thin film equation Fourier pseudo-spectral approximation the scalar auxiliary variable(SAV)method Crank-Nicolson temporal discretization energy stability optimal rate convergence analysis. 

分 类 号：O17[理学—数学]