doi: 10.3934/dcdsb.2020311

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

Received  March 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by NSFC grant 11771213, 61872422

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.

Citation: Lin Lu, Qi Wang, Yongzhong Song, Yushun Wang. Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020311
References:
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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.  Google Scholar

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L. BrugnanoF. 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.   Google Scholar

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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.  Google Scholar

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J. CaiJ. HongY. 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.  Google Scholar

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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.  Google Scholar

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J. CaiY. 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.  Google Scholar

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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.  Google Scholar

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J. CaiY. 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.  Google Scholar

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E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealeB. 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.  Google Scholar

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A. ChristliebJ. JonesK. PromislowB. 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.  Google Scholar

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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.  Google Scholar

[22]

Q. HongY. 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.  Google Scholar

[23]

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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.  Google Scholar

[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.  Google Scholar

[26]

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[27]

Z. MuY. GongW. 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.  Google Scholar

[28]

Z. QiaoZ. 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.  Google Scholar

[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.  Google Scholar

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J. ShenC. WangX. 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.  Google Scholar

[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.  Google Scholar

[32]

J. ShenJ. 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.  Google Scholar

[33]

J. ShenX. YangB. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Y. Wang and J. Hong, Multi-symplectic algorithms for Hamiltonian partial differential equations, Commun. Appl. Math. Comput, 27 (2013), 163-230.   Google Scholar

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Y. WangB. 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.  Google Scholar

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C. WangX. 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.  Google Scholar

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A. Willoughby and P. Capper, Molecular Beam Epitaxy: Materials and Applications for Electronics and Optoelectronics, Springer, 2019. Google Scholar

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S. M. WiseC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

X. YangJ. 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.  Google Scholar

[45]

J. ZhaoQ. 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.  Google Scholar

show all references

References:
[1]

T. J. Bridges, Multi-symplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190.  doi: 10.1017/S0305004196001429.  Google Scholar

[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.  Google Scholar

[3]

L. BrugnanoF. 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.   Google Scholar

[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.  Google Scholar

[5]

J. CaiJ. HongY. 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.  Google Scholar

[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.  Google Scholar

[7]

J. CaiY. 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.  Google Scholar

[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.  Google Scholar

[9]

J. CaiY. 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.  Google Scholar

[10]

E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealeB. 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.  Google Scholar

[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.  Google Scholar

[12]

Q. ChengJ. 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.  Google Scholar

[13]

A. ChristliebJ. JonesK. PromislowB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. GongJ. 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.  Google Scholar

[18]

Z. GuanJ. S. LowengrubC. 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.  Google Scholar

[19]

M. Guina and S. M. Wang, Molecular Beam Epitaxy, Elsevier, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Q. HongY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. E. MarsdenG. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys., 199 (1998), 351-395.  doi: 10.1007/s002200050505.  Google Scholar

[27]

Z. MuY. GongW. 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.  Google Scholar

[28]

Z. QiaoZ. 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.  Google Scholar

[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.  Google Scholar

[30]

J. ShenC. WangX. 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.  Google Scholar

[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.  Google Scholar

[32]

J. ShenJ. 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.  Google Scholar

[33]

J. ShenX. YangB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. Wang and J. Hong, Multi-symplectic algorithms for Hamiltonian partial differential equations, Commun. Appl. Math. Comput, 27 (2013), 163-230.   Google Scholar

[37]

Y. WangB. 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.  Google Scholar

[38]

C. WangX. 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.  Google Scholar

[39]

A. Willoughby and P. Capper, Molecular Beam Epitaxy: Materials and Applications for Electronics and Optoelectronics, Springer, 2019. Google Scholar

[40]

S. M. WiseC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

X. YangJ. 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.  Google Scholar

[45]

J. ZhaoQ. 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.  Google Scholar

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 $
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
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
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 $
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 $
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
Figure 7.  The Energy for LEDP-I and LEDP-II via different time steps
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} }) $
Table 1.  Mesh refinement test for LEDP-I at $ t = 1 $
$ N $ $ \tau $ Error Order CPU time
$ L^\infty $ error $ L^{2} $ error $ L^\infty $ order $ L^{2} $ order
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
$ N $ $ \tau $ Error Order CPU time
$ L^\infty $ error $ L^{2} $ error $ L^\infty $ order $ L^{2} $ order
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
Table 2.  Mesh refinement test for LEDP-II at $ t = 1 $
$ N $ $ \tau $ Error Order CPU time
$ L^\infty $ error $ L^{2} $ error $ L^\infty $ order $ L^{2} $ order
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
$ N $ $ \tau $ Error Order CPU time
$ L^\infty $ error $ L^{2} $ error $ L^\infty $ order $ L^{2} $ order
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
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