Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

Lin Lu; Qi Wang; Yongzhong Song; Yushun Wang

doi:10.3934/dcdsb.2020311

Article Contents

2021, Volume 26, Issue 9: 4745-4765. Doi: 10.3934/dcdsb.2020311

This issue Previous Article Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications Next Article Invasion dynamics of a diffusive pioneer-climax model: Monotone and non-monotone cases

Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection

1.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China
2.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

^* Corresponding author: Yushun Wang
^* Corresponding author: Yushun Wang

Received: March 2020

Revised: August 2020

Early access: October 2020

Published: September 2021

The first author is supported by NSFC grant 11771213, 61872422

Abstract Full Text(HTML) Figure(8) / Table(2) Related Papers Cited by

Abstract

Based on the local energy dissipation property of the molecular beam epitaxy (MBE) model with slope selection, we develop three, second order fully discrete, local energy dissipation rate preserving (LEDP) algorithms for the model using finite difference methods. For periodic boundary conditions, we show that these algorithms are global energy dissipation rate preserving (GEDP). For adiabatic, physical boundary conditions, we construct two GEDP algorithms from the three LEDP ones with consistently discretized physical boundary conditions. In addition, we show that all the algorithms preserve the total mass at the discrete level as well. Mesh refinement tests are conducted to confirm the convergence rates of the algorithms and two benchmark examples are presented to show the accuracy and performance of the methods.

Keywords:

Mathematics Subject Classification: Primary: 65M06; Secondary: 80M20.

Citation:

Full Text(HTML)

Figure 1. The isolines of numerical solutions of $ \phi $ in Example 2 using LEDP-I and LEDP-II, respectively. (a-f) are obtained from LEDP-I while (g-l) from LEDP-II. Snapshots are taken at $ t = 0, 0.05, 2.5, 5.5, 8, 30 $, respectively. The time step is set as $ \tau = 1.0e-3 $

Download: Full-size image PowerPoint slide

Figure 2. Time evolution of the error in mass and global energy with $ N = 129 $ and $ \tau = 1.0e-3 $ in Example 2 using LEDP-I and LEDP-II, respectively

Download: Full-size image PowerPoint slide

Figure 3. Time evolution of energy and maximal residue of the local energy dissipation law with $ N = 129 $, $ \tau = 1.0e-3 $ and $ \tau $ based on adaptive time stepping algorithm in Example 2 using LEDP-I, LEDP-II, respectively

Download: Full-size image PowerPoint slide

Figure 4. The isolines of numerical solutions of $ \phi $ (left) and its Laplacian $ \Delta \phi $ (right) in Example 3 using LEDP-I. Snapshots are taken at $ t = 0, 5, 10, 20, 40, 80 $. The time and space step are set as $ \tau = 1.0e-3 $ and $ N = 513 $

Download: Full-size image PowerPoint slide

Figure 5. The isolines of numerical solutions of the $ \phi $ (left) and its Laplacian $ \Delta \phi $ (right) in Example 3 using LEDP-II. Snapshots are taken at $ t = 0, 5, 10, 20, 40, 80 $. The time and space step are set as $ \tau = 1.0e-3 $ and $ N = 513 $

Download: Full-size image PowerPoint slide

Figure 6. Time evolution of the error in mass, energy and maximal residue with $ N = 513 $ and $ \tau = 1.0e-3 $ in Example 3 using LEDP-I and LEDP-II, respectively

Download: Full-size image PowerPoint slide

Figure 7. The Energy for LEDP-I and LEDP-II via different time steps

Download: Full-size image PowerPoint slide

Figure 8. The numerical results show the proper power law behavior in the decaying energy as $ O(t^{-\frac{1}{3} }) $ and roughness as $ O(t^{\frac{1}{3} }) $

Download: Full-size image PowerPoint slide

Table 1. Mesh refinement test for LEDP-I at $ t = 1 $

$ N $	$ \tau $	Error		Order		CPU time
$ N $	$ \tau $	$ L^\infty $ error	$ L^{2} $ error	$ L^\infty $ order	$ L^{2} $ order	CPU time
11	0.1	0.1805	0.5671	–	–	6.24e-1
33	1/30	0.0170	0.0535	2.1495	2.1495	8.71e-1
99	1/90	0.0019	0.0058	2.0160	2.0160	4.82
297	1/270	2.0605e-4	6.4733e-4	2.0018	2.0018	5.75e+1
891	1/810	2.2890e-5	7.1910e-5	2.0002	2.0002	7.37e+2

| Show Table

DownLoad: CSV

Table 2. Mesh refinement test for LEDP-II at $ t = 1 $

$ N $	$ \tau $	Error		Order		CPU time
$ N $	$ \tau $	$ L^\infty $ error	$ L^{2} $ error	$ L^\infty $ order	$ L^{2} $ order	CPU time
11	0.1	1.9180e-4	6.0195e-4	–	–	1.25e-1
33	1/30	2.1309e-5	6.6866e-5	2.0001	2.0002	2.47e-1
99	1/90	2.3678e-6	7.4296e-6	1.9999	2.0000	2.01
297	1/270	2.6376e-7	8.2575e-7	1.9977	1.9997	1.16e+1
891	1/810	2.9864e-8	9.1995e-8	1.9829	1.9976	8.79e+1

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	T. J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190. doi: 10.1017/S0305004196001429.
[2]	T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, 39 (2006), 5287-5320. doi: 10.1088/0305-4470/39/19/S02.
[3]	L. Brugnano, F. Iavernaro and D. Trigiante, Hamiltonian boundary value methods (energy preserving discrete line integral methods), JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5 (2010), 17-37.
[4]	L. Brugnano and Y. Sun, Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems, Numer. Algorithms, 65 (2014), 611-632. doi: 10.1007/s11075-013-9769-9.
[5]	J. Cai, J. Hong, Y. Wang and Y. Gong, Two energy-conserved splitting methods for three-dimensional time-domain Maxwell's equations and the convergence analysis, SIAM J. Numer. Anal., 53 (2015), 1918-1940. doi: 10.1137/140971609.
[6]	J. Cai and J. Shen, Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 401 (2020), 108975, 17 pp. doi: 10.1016/j.jcp.2019.108975.
[7]	J. Cai, Y. Wang and H. Liang, Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system, J. Comput. Phys., 239 (2013), 30-50. doi: 10.1016/j.jcp.2012.12.036.
[8]	J. Cai and Y. Wang, Local structure-preserving algorithms for the "good" Boussinesq equation, J. Comput. Phys., 239 (2013), 72-89. doi: 10.1016/j.jcp.2013.01.009.
[9]	J. Cai, Y. Wang and C. Jiang, Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs, Comput. Phys. Comm., 235 (2019), 210-220. doi: 10.1016/j.cpc.2018.08.015.
[10]	E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neale, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789. doi: 10.1016/j.jcp.2012.06.022.
[11]	Q. Cheng, C. Liu and J. Shen, A new lagrange multiplier approach for gradient flows, Comput. Methods Appl. Mech. Engrg., 367 (2020), 113070, 20 pp. doi: 10.1016/j.cma.2020.113070.
[12]	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.
[13]	A. Christlieb, J. Jones, K. Promislow, B. Wetton and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, J. Comput. Phys., 257 (2014), 193-215. doi: 10.1016/j.jcp.2013.09.049.
[14]	N. Del Buono and C. Mastroserio, Explicit methods based on a class of four stage fourth order Runge–Kutta methods for preserving quadratic laws, J. Comput. Appl. Math., 140 (2002), 231-243. doi: 10.1016/S0377-0427(01)00398-3.
[15]	M. Doi, Onsager's variational principle in soft matter, J. Phys.: Condens. Matter, 23 (2011), 284118. doi: 10.1088/0953-8984/23/28/284118.
[16]	D. Furihata, Finite difference schemes for $\partial u/\partial t = (\partial/\partial x)^\alpha\delta G/\delta u$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377.
[17]	Y. Gong, J. Cai and Y. Wang, Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), 80-102. doi: 10.1016/j.jcp.2014.09.001.
[18]	Z. Guan, J. S. Lowengrub, C. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001.
[19]	M. Guina and S. M. Wang, Molecular Beam Epitaxy, Elsevier, 2013.
[20]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer, Heidelberg, 2010.
[21]	Q. Hong, J. Li and Q. Wang, Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations, Appl. Math. Lett., 110 (2020), 106576, 9 pp. doi: 10.1016/j.aml.2020.106576.
[22]	Q. Hong, Y. Wang and Y. Gong, Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation, Numer. Methods Partial Differential Equations, 36 (2020), 394-417. doi: 10.1002/num.22434.
[23]	L. Huang, Z. Tian and Y. Cai, Compact local structure-preserving algorithms for the nonlinear Schrödinger equation with wave operator, Math. Probl. Eng., 2020 (2020), 4345278, 12 pp. doi: 10.1155/2020/4345278.
[24]	B. Li and J. Liu, Thin film epitaxy with or without slope selection, European J. Appl. Math., 14 (2003), 713-743. doi: 10.1017/S095679250300528X.
[25]	Y.-W. Li and X. Wu, Functionally fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems, SIAM J. Numer. Anal., 54 (2016), 2036-2059. doi: 10.1137/15M1032752.
[26]	J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505.
[27]	Z. Mu, Y. Gong, W. Cai and Y. Wang, Efficient local energy dissipation preserving algorithms for the Cahn-Hilliard equation, J. Comput. Phys., 374 (2018), 654-667. doi: 10.1016/j.jcp.2018.08.004.
[28]	Z. Qiao, Z. Zhang and T. Tang, An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), 1395-1414. doi: 10.1137/100812781.
[29]	S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), 473-499. doi: 10.1006/jcph.1999.6372.
[30]	J. Shen, C. Wang, X. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Enrich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125. doi: 10.1137/110822839.
[31]	J. Shen and J. Xu, Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows, SIAM J. Numer. Anal., 56 (2018), 2895-2912. doi: 10.1137/17M1159968.
[32]	J. Shen, J. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416. doi: 10.1016/j.jcp.2017.10.021.
[33]	J. Shen, X. Yang, B. Wetton and M. Willoughby, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Disc. Conti. Dyn. Syst. Ser. A, 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.
[34]	S. Sun, J. Li, J. Zhao and Q. Wang, Structure-preserving numerical approximations to a non-isothermal hydrodynamic model of binary fluid flows, J. Sci. Comput., 83 (2020), 50, 43 pp. doi: 10.1007/s10915-020-01229-6.
[35]	W. Tang and Y. Sun, Time finite element methods: A unified framework for numerical discretizations of ODEs, Appl. Math. Comput., 219 (2012), 2158-2179. doi: 10.1016/j.amc.2012.08.062.
[36]	Y. Wang and J. Hong, Multi-symplectic algorithms for Hamiltonian partial differential equations, Commun. Appl. Math. Comput, 27 (2013), 163-230.
[37]	Y. Wang, B. Wang and M. Qin, Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A, 51 (2008), 2115-2136. doi: 10.1007/s11425-008-0046-7.
[38]	C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst., 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.
[39]	A. Willoughby and P. Capper, Molecular Beam Epitaxy: Materials and Applications for Electronics and Optoelectronics, Springer, 2019.
[40]	S. M. Wise, C. Wang and J. S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47 (2009), 2269-2288. doi: 10.1137/080738143.
[41]	X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Disc. Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070. doi: 10.3934/dcdsb.2009.11.1057.
[42]	X.-G. Yang, M. G. Forest and Q. Wang, Near equilibrium dynamics and one-dimensional spatial-temporal structures of polar active liquid crystals, Chin. Phys. B, 23 (2014), 118701. doi: 10.1088/1674-1056/23/11/118701.
[43]	X. Yang, J. Li, M. G. Forest and Q. Wang, Hydrodynamic theories for flows of active liquid crystals and the generalized Onsager principle, Entropy, 18 (2016), 202, 28 pp. doi: 10.3390/e18060202.
[44]	X. Yang, J. Zhao and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys., 333 (2017), 104-127. doi: 10.1016/j.jcp.2016.12.025.
[45]	J. Zhao, Q. Wang and X. Yang, Numerical approximations for a phase field dendritic crystal growth model based on invariant energy quadratization approach, Internat. J. Numer. Methods Engrg., 110 (2017), 279-300. doi: 10.1002/nme.5372.