Discretisation and Numerical Tests of a Diffuse-Interface Model with Ehrlich-Schwoebel Barrier
Félix Otto, Patrick Penzler, Tobias Rump · Birkhäuser-Verlag eBooks · 2005
We consider a step flow model for epitaxial growth, as proposed by Burton, Cabrera and Frank [3]. This type of model is discrete in the growth direction but continuous in the lateral directions. The effect of the Ehrlich-Schwoebel barrier, which limits the attachment rate of adatoms to a step from an upper terrace, is included. Mathematically, this model is a dynamic free boundary problem for the steps. In [6], we proposed a diffuse-interface approximation which reproduces an arbitrary Ehrlich-Schwoebel barrier. It is a version of the Cahn-Hilliard equation with variable mobility.In this paper, we propose a discretisation for this diffuse-interface approximation. Our approach is guided by the fact that the diffuse-interface approximation has a conserved quantity and a Liapunov functional. We are lead to an implicit finite volume discretisation of symmetric structure.We test the discretisation by comparison with the matched asymptotic analysis carried out in [6]. We also test the diffuse-interface approximation itself by comparison with theoretically known features of the original free boundary problem. More precisely, we investigate quantitatively the phenomena of step bunching and the Bales-Zangwill instability.