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Energetics and coarsening analysis of a simplified non-linear surface growth model
1. | ENSA, LIPIM laboratory, University Sultan Moulay Slimane, Khouribga, Morocco |
2. | FST, LAMAI Laboratory, University Cadi Ayyad, Marrakesh, Morocco |
$ u $ |
$ E(u) = \int_{\Omega} \left[ \frac{1}{2} \ln\left(1+\left|\nabla u \right|^2\right) - \left|\nabla u \right| \arctan\left(\left|\nabla u \right|\right) + \frac{1}{2} \left|\Delta u \right|^2 \right] {\rm d}x, $ |
$ \Omega $ |
$ \left|\Delta u \right|^2 $ |
$ \left( \left|\nabla u \right| \right) $ |
$ \left| \Omega \right| $ |
References:
[1] |
W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise,
A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.
doi: 10.1007/s10915-011-9559-2. |
[2] |
W. Chen, C. Wang, X. Wang and S. M. Wise,
A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.
doi: 10.1007/s10915-013-9774-0. |
[3] |
L. Chen, J. Zhao and X. Yang,
Regularized linear schemes for the molecular beam epitaxy model with slope selection, Appl. Numer. Math., 128 (2018), 139-156.
doi: 10.1016/j.apnum.2018.02.004. |
[4] |
K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154–185. arXiv: 1903.03296.
doi: 10.1007/s10915-019-01008-y. |
[5] |
Q. Cheng, J. Shen and X. Yang,
Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach, J. Sci. Comput., 78 (2019), 1467-1487.
doi: 10.1007/s10915-018-0832-5. |
[6] |
Z. Csahók, C. Misbah and A. Valance,
A class of nonlinear front evolution equations derived from geometry and conservation, Phys. D, 128 (1999), 87-100.
doi: 10.1016/S0167-2789(98)00320-0. |
[7] |
L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection, Phys. Rev. Lett., 78 (1997), 90.
doi: 10.1103/PhysRevLett.78.90. |
[8] |
M. Grasselli, G. Mola and A. Yagi,
On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004.
|
[9] |
M. Guedda and H. Trojette,
Coarsening in an interfacial equation without slope selection revisited: Analytical results, Phys. Lett. A, 374 (2010), 4308-4311.
doi: 10.1016/j.physleta.2010.08.052. |
[10] |
L. Ju, X. Li, Z. Qiao and H. Zhang,
Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comp., 87 (2018), 1859-1885.
doi: 10.1090/mcom/3262. |
[11] |
H. Khalfi, N. E. Alaa and M. Guedda,
Coarsening properties of a nonlinear front evolution equation, J. Adv. Math. Stud., 11 (2018), 295-305.
|
[12] |
H. Khalfi, N. E. Alaa and M. Guedda,
Period steady-state identification for a nonlinear front evolution equation using genetic algorithms, International Journal of Bio-Inspired Computation, 12 (2018), 196-202.
doi: 10.1504/IJBIC.2018.094647. |
[13] |
H. Khalfi, M. Pierre, N. E. Alaa and M. Guedda,
Convergence to equilibrium of a DC algorithm for an epitaxial growth model., Int. J. Numer. Anal. Model., 16 (2019), 98-411.
|
[14] |
H. G. Lee, J. Shin and J.-Y. Lee,
A second-order operator splitting fourier spectral method for models of epitaxial thin film growth, J. Sci. Comput., 71 (2017), 1303-1318.
doi: 10.1007/s10915-016-0340-4. |
[15] |
W. Li, W. Chen, C. Wang, Y. Yan and R. He,
A second order energy stable linear scheme for a thin film model without slope selection, J. Sci. Comput., 76 (2018), 1905-1937.
doi: 10.1007/s10915-018-0693-y. |
[16] |
B. Li and J.-G. Liu,
Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[17] |
B. Li and J.-G. Liu,
Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[18] |
C. Misbah, O. Pierre-Louis and Y. Saito, Crystal surfaces in and out of equilibrium: A modern view, Reviews of Modern Physics, 82 (2010), 981.
doi: 10.1103/RevModPhys.82.981. |
[19] |
O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug and P. Politi, New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces, Physical Review Letters, 80 (1998), 4221.
doi: 10.1103/PhysRevLett.80.4221. |
[20] |
P. Politi and J. Villain, Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model, Physical Review B, 54 (1996), 5114.
doi: 10.1103/PhysRevB.54.5114. |
[21] |
P. Politi and C. Misbah, Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law, Phys. Rev. E, 73 (2006), 036133, 15 pp.
doi: 10.1103/PhysRevE.73.036133. |
[22] |
Z. Qiao, Z.-Z. Sun and Z. Zhang,
Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection, Math. Comp., 84 (2015), 653-674.
doi: 10.1090/S0025-5718-2014-02874-3. |
[23] |
M. Rost, P. Šmilauer and J. Krug,
Unstable epitaxy on vicinal surfaces, Surface Science, 369 (1996), 393-402.
doi: 10.1016/S0039-6028(96)00905-3. |
[24] |
R. Schwoebel,
Step motion on crystal surfaces. II, Journal of Applied Physics, 40 (1969), 614-618.
doi: 10.1063/1.1657442. |
show all references
References:
[1] |
W. Chen, S. Conde, C. Wang, X. Wang and S. M. Wise,
A linear energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 52 (2012), 546-562.
doi: 10.1007/s10915-011-9559-2. |
[2] |
W. Chen, C. Wang, X. Wang and S. M. Wise,
A linear iteration algorithm for a second-order energy stable scheme for a thin film model without slope selection, J. Sci. Comput., 59 (2014), 574-601.
doi: 10.1007/s10915-013-9774-0. |
[3] |
L. Chen, J. Zhao and X. Yang,
Regularized linear schemes for the molecular beam epitaxy model with slope selection, Appl. Numer. Math., 128 (2018), 139-156.
doi: 10.1016/j.apnum.2018.02.004. |
[4] |
K. Cheng, Z. Qiao and C. Wang, A third order exponential time differencing numerical scheme for no-slope-selection epitaxial thin film model with energy stability, J. Sci. Comput., 81 (2019), 154–185. arXiv: 1903.03296.
doi: 10.1007/s10915-019-01008-y. |
[5] |
Q. Cheng, J. Shen and X. Yang,
Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach, J. Sci. Comput., 78 (2019), 1467-1487.
doi: 10.1007/s10915-018-0832-5. |
[6] |
Z. Csahók, C. Misbah and A. Valance,
A class of nonlinear front evolution equations derived from geometry and conservation, Phys. D, 128 (1999), 87-100.
doi: 10.1016/S0167-2789(98)00320-0. |
[7] |
L. Golubović, Interfacial coarsening in epitaxial growth models without slope selection, Phys. Rev. Lett., 78 (1997), 90.
doi: 10.1103/PhysRevLett.78.90. |
[8] |
M. Grasselli, G. Mola and A. Yagi,
On the longtime behavior of solutions to a model for epitaxial growth, Osaka J. Math., 48 (2011), 987-1004.
|
[9] |
M. Guedda and H. Trojette,
Coarsening in an interfacial equation without slope selection revisited: Analytical results, Phys. Lett. A, 374 (2010), 4308-4311.
doi: 10.1016/j.physleta.2010.08.052. |
[10] |
L. Ju, X. Li, Z. Qiao and H. Zhang,
Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comp., 87 (2018), 1859-1885.
doi: 10.1090/mcom/3262. |
[11] |
H. Khalfi, N. E. Alaa and M. Guedda,
Coarsening properties of a nonlinear front evolution equation, J. Adv. Math. Stud., 11 (2018), 295-305.
|
[12] |
H. Khalfi, N. E. Alaa and M. Guedda,
Period steady-state identification for a nonlinear front evolution equation using genetic algorithms, International Journal of Bio-Inspired Computation, 12 (2018), 196-202.
doi: 10.1504/IJBIC.2018.094647. |
[13] |
H. Khalfi, M. Pierre, N. E. Alaa and M. Guedda,
Convergence to equilibrium of a DC algorithm for an epitaxial growth model., Int. J. Numer. Anal. Model., 16 (2019), 98-411.
|
[14] |
H. G. Lee, J. Shin and J.-Y. Lee,
A second-order operator splitting fourier spectral method for models of epitaxial thin film growth, J. Sci. Comput., 71 (2017), 1303-1318.
doi: 10.1007/s10915-016-0340-4. |
[15] |
W. Li, W. Chen, C. Wang, Y. Yan and R. He,
A second order energy stable linear scheme for a thin film model without slope selection, J. Sci. Comput., 76 (2018), 1905-1937.
doi: 10.1007/s10915-018-0693-y. |
[16] |
B. Li and J.-G. Liu,
Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743.
doi: 10.1017/S095679250300528X. |
[17] |
B. Li and J.-G. Liu,
Epitaxial growth without slope selection: Energetics, coarsening, and dynamic scaling, J. Nonlinear Sci., 14 (2004), 429-451.
doi: 10.1007/s00332-004-0634-9. |
[18] |
C. Misbah, O. Pierre-Louis and Y. Saito, Crystal surfaces in and out of equilibrium: A modern view, Reviews of Modern Physics, 82 (2010), 981.
doi: 10.1103/RevModPhys.82.981. |
[19] |
O. Pierre-Louis, C. Misbah, Y. Saito, J. Krug and P. Politi, New nonlinear evolution equation for steps during molecular beam epitaxy on vicinal surfaces, Physical Review Letters, 80 (1998), 4221.
doi: 10.1103/PhysRevLett.80.4221. |
[20] |
P. Politi and J. Villain, Ehrlich-Schwoebel instability in molecular-beam epitaxy: A minimal model, Physical Review B, 54 (1996), 5114.
doi: 10.1103/PhysRevB.54.5114. |
[21] |
P. Politi and C. Misbah, Nonlinear dynamics in one dimension: A criterion for coarsening and its temporal law, Phys. Rev. E, 73 (2006), 036133, 15 pp.
doi: 10.1103/PhysRevE.73.036133. |
[22] |
Z. Qiao, Z.-Z. Sun and Z. Zhang,
Stability and convergence of second-order schemes for the nonlinear epitaxial growth model without slope selection, Math. Comp., 84 (2015), 653-674.
doi: 10.1090/S0025-5718-2014-02874-3. |
[23] |
M. Rost, P. Šmilauer and J. Krug,
Unstable epitaxy on vicinal surfaces, Surface Science, 369 (1996), 393-402.
doi: 10.1016/S0039-6028(96)00905-3. |
[24] |
R. Schwoebel,
Step motion on crystal surfaces. II, Journal of Applied Physics, 40 (1969), 614-618.
doi: 10.1063/1.1657442. |









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