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March  2010, 28(1): 405-423. doi: 10.3934/dcds.2010.28.405

Unconditionally stable schemes for equations of thin film epitaxy

Cheng Wang 1, , Xiaoming Wang 2, and  Steven M. Wise 3,

1. 

Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747-2300, United States
2. 

Department of Mathematics, The Florida State University, Tallahassee, FL 32306-4510
3. 

Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-0612, United States

Received  April 2009 Revised  November 2009 Published  April 2010

We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|2-1)2) and without slope selection (F(y)= -1/2ln(1+|y|2)). We conclude the paper with some preliminary computations that employ the proposed schemes.
Keywords: energy stability, long-time stability, Epitaxial growth, convexity splitting..
Mathematics Subject Classification: Primary: 65M12; Secondary: 37L6.
Citation: Cheng Wang, Xiaoming Wang, Steven M. Wise. Unconditionally stable schemes for equations of thin film epitaxy. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405
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