A compact exponential method for the efficient numerical simulation of the dewetting process of viscous thin films
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In this work, we propose a discrete mathematical system to model the evolution of the thickness of two-dimensional viscous thin films subject to a dewetting process. The continuous model under consideration is a degenerate partial differential equation that generalizes the classical thin film equation, and considers the inclusion of a singular potential. The analytical model is discretized using an exponential method that is capable of preserving the positive character of the approximations. In addition, the explicit nature of our approach results in an economic computer implementation which produces fast simulations. We provide some illustrative examples on the dynamics of the growth of thin films in the presence/absence of a dewetting process. The qualitative results exhibit the appearance of typical patterns obtained in experimental settings. The technique was validated against Bhattacharya’s method and a standard explicit discretization of the mathematical model.
KeywordsDegenerate thin-film equation Compact exponential method Preservation of positivity Computationally efficient numerical technique
Mathematics Subject Classification92-08 65M06 92C37 35K55
Beforehand, we want to thank the anonymous reviewers and the associate editor in charge of handling this manuscript, for all the invaluable suggestions and comments. Also, the authors would like to thank the Dean of the Faculty of Sciences of the Universidad Autónoma de Aguascalientes (UAA) for the partial financial funding of this work. The second author wishes to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT) through the Award No. 260373. Finally, we acknowledge the technical assistance of Dr. Diana E. García-Rodriguez, who was in charge of obtaining the atomic force microscopy images used in this work, and Mr. Axel Guzmán-Chávez, who coded a raw version of our method as part of an undergraduate research project at UAA.
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