Unconditionally stable schemes for equations of thin film epitaxy

Cheng Wang; Xiaoming Wang; Steven M. Wise

doi:10.3934/dcds.2010.28.405

Article Contents

2010, Volume 28, Issue 1: 405-423. Doi: 10.3934/dcds.2010.28.405

This issue Previous Article Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit Next Article

Unconditionally stable schemes for equations of thin film epitaxy

1.
Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747-2300
2.
Department of Mathematics, The Florida State University, Tallahassee, FL 32306-4510
3.
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-0612

Received: March 31, 2009

Revised: October 31, 2009

Published: February 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 65M12; Secondary: 37L65.

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