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:
[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

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
[1]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[2]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[3]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[4]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[5]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[6]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[7]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[8]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[10]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[11]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[12]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[13]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[15]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[16]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[17]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[18]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[19]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[20]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (21)
  • HTML views (53)
  • Cited by (0)

Other articles
by authors

[Back to Top]