American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405
[1]

Amjad Khan, Dmitry E. Pelinovsky. Long-time stability of small FPU solitary waves. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2065-2075. doi: 10.3934/dcds.2017088
[2]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[3]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112
[4]

Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170
[5]

Manuel Núñez. The long-time evolution of mean field magnetohydrodynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 465-478. doi: 10.3934/dcdsb.2004.4.465
[6]

O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete & Continuous Dynamical Systems, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131
[7]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
[8]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[9]

Yang Liu. Long-time behavior of a class of viscoelastic plate equations. Electronic Research Archive, 2020, 28 (1) : 311-326. doi: 10.3934/era.2020018
[10]

A. Kh. Khanmamedov. Long-time behaviour of doubly nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1373-1400. doi: 10.3934/cpaa.2009.8.1373
[11]

Igor Chueshov, Stanislav Kolbasin. Long-time dynamics in plate models with strong nonlinear damping. Communications on Pure & Applied Analysis, 2012, 11 (2) : 659-674. doi: 10.3934/cpaa.2012.11.659
[12]

Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023
[13]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[14]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[15]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[16]

H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119
[17]

Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098
[18]

Yuguo Lin, Daqing Jiang. Long-time behaviour of a perturbed SIR model by white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1873-1887. doi: 10.3934/dcdsb.2013.18.1873
[19]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[20]

C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (86)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy