Close Search Advanced
Journals
Resources
About Us
Open Access

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model
Preview
Add to basket

Year:    2021

Author:    Yonghong Hao, Qiumei Huang, Cheng Wang

Communications in Computational Physics, Vol. 29 (2021), Iss. 3 : pp. 905–929

Abstract

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $−A∆t^2∆^2_N (u^{n+1}−u^n)$, is added in the numerical scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $ℓ^∞(0,T;ℓ^2)∩ℓ^2(0,T;H^2_h)$ norm, with the help of a linearized estimate for the nonlinear error terms. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $ε = 0.02$ (up to $T = 3×10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0074

Communications in Computational Physics, Vol. 29 (2021), Iss. 3 : pp. 905–929

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    25

Keywords:    Epitaxial thin film growth no-slope-selection third order backward differentiation formula energy stability optimal rate convergence analysis.

Author Details

Yonghong Hao

Qiumei Huang

Cheng Wang

  1. Unconditionally stable exponential time differencing schemes for the mass‐conserving Allen–Cahn equation with nonlocal and local effects

    Jiang, Kun | Ju, Lili | Li, Jingwei | Li, Xiao

    Numerical Methods for Partial Differential Equations, Vol. 38 (2022), Iss. 6 P.1636

    https://doi.org/10.1002/num.22827 [Citations: 12]

  2. A linear second-order in time unconditionally energy stable finite element scheme for a Cahn–Hilliard phase-field model for two-phase incompressible flow of variable densities

    Fu, Guosheng | Han, Daozhi

    Computer Methods in Applied Mechanics and Engineering, Vol. 387 (2021), Iss. P.114186

    https://doi.org/10.1016/j.cma.2021.114186 [Citations: 7]

  3. L 2 norm error estimates of BDF methods up to fifth-order for the phase field crystal model

    Liao, Hong-lin | Kang, Yuanyuan

    IMA Journal of Numerical Analysis, Vol. 44 (2024), Iss. 4 P.2138

    https://doi.org/10.1093/imanum/drad047 [Citations: 5]

  4. An Exponential Time Differencing Runge–Kutta Method ETDRK32 for Phase Field Models

    Cao, Weichen | Yang, Hengli | Chen, Wenbin

    Journal of Scientific Computing, Vol. 99 (2024), Iss. 1

    https://doi.org/10.1007/s10915-024-02474-9 [Citations: 0]

  5. Constructing time integration with controllable errors for constrained mechanical systems

    Xu, Xiaoming | Chen, Yanghui | Luo, Jiahui | Liu, Jiafu | Peng, Haijun | Wu, Zhigang

    Applied Mathematical Modelling, Vol. 118 (2023), Iss. P.185

    https://doi.org/10.1016/j.apm.2023.01.037 [Citations: 3]

  6. Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

    Hong, Qi | Gong, Yuezheng | Zhao, Jia | Wang, Qi

    Applied Numerical Mathematics, Vol. 170 (2021), Iss. P.321

    https://doi.org/10.1016/j.apnum.2021.08.002 [Citations: 8]

  7. A Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Square Phase Field Crystal Equation

    Wang, Min | Huang, Qiumei | Wang, Cheng

    Journal of Scientific Computing, Vol. 88 (2021), Iss. 2

    https://doi.org/10.1007/s10915-021-01487-y [Citations: 70]

  8. A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility

    Hou, Dianming | Ju, Lili | Qiao, Zhonghua

    Mathematics of Computation, Vol. 92 (2023), Iss. 344 P.2515

    https://doi.org/10.1090/mcom/3843 [Citations: 8]

  9. The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

    Meng, Xiaoqing | Cheng, Aijie | Liu, Zhengguang

    Communications in Nonlinear Science and Numerical Simulation, Vol. 137 (2024), Iss. P.108169

    https://doi.org/10.1016/j.cnsns.2024.108169 [Citations: 1]

  10. A decoupled, unconditionally energy-stable and structure-preserving finite element scheme for the incompressible MHD equations with magnetic-electric formulation

    Zhang, Xiaodi | Su, Haiyan

    Computers & Mathematics with Applications, Vol. 146 (2023), Iss. P.45

    https://doi.org/10.1016/j.camwa.2023.06.029 [Citations: 0]

  11. Phase field modeling and computation of multi-component droplet evaporation

    Yang, Junxiang

    Computer Methods in Applied Mechanics and Engineering, Vol. 401 (2022), Iss. P.115675

    https://doi.org/10.1016/j.cma.2022.115675 [Citations: 7]

  12. Motion by Mean Curvature with Constraints Using a Modified Allen–Cahn Equation

    Kwak, Soobin | Lee, Hyun Geun | Li, Yibao | Yang, Junxiang | Lee, Chaeyoung | Kim, Hyundong | Kang, Seungyoon | Kim, Junseok

    Journal of Scientific Computing, Vol. 92 (2022), Iss. 1

    https://doi.org/10.1007/s10915-022-01862-3 [Citations: 3]

  13. A general class of linear unconditionally energy stable schemes for the gradient flows

    Tan, Zengqiang | Tang, Huazhong

    Journal of Computational Physics, Vol. 464 (2022), Iss. P.111372

    https://doi.org/10.1016/j.jcp.2022.111372 [Citations: 4]

  14. An energy stable linear BDF2 scheme with variable time-steps for the molecular beam epitaxial model without slope selection

    Kang, Yuanyuan | Liao, Hong-lin | Wang, Jindi

    Communications in Nonlinear Science and Numerical Simulation, Vol. 118 (2023), Iss. P.107047

    https://doi.org/10.1016/j.cnsns.2022.107047 [Citations: 3]

  15. A Second-Order Exponential Time Differencing Multi-step Energy Stable Scheme for Swift–Hohenberg Equation with Quadratic–Cubic Nonlinear Term

    Cui, Ming | Niu, Yiyi | Xu, Zhen

    Journal of Scientific Computing, Vol. 99 (2024), Iss. 1

    https://doi.org/10.1007/s10915-024-02490-9 [Citations: 1]

  16. A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

    Sun, Jingwei | Zhang, Hong | Qian, Xu | Song, Songhe

    Journal of Computational Physics, Vol. 492 (2023), Iss. P.112414

    https://doi.org/10.1016/j.jcp.2023.112414 [Citations: 5]

  17. Thermal-fluid topology optimization with unconditional energy stability and second-order accuracy via phase-field model

    Xia, Qing | Sun, Gangming | Yu, Qian | Kim, Junseok | Li, Yibao

    Communications in Nonlinear Science and Numerical Simulation, Vol. 116 (2023), Iss. P.106782

    https://doi.org/10.1016/j.cnsns.2022.106782 [Citations: 20]

  18. A variational Crank–Nicolson ensemble Monte Carlo algorithm for a heat equation under uncertainty

    Ye, Changlun | Yao, Tingfu | Bi, Hai | Luo, Xianbing

    Journal of Computational and Applied Mathematics, Vol. 451 (2024), Iss. P.116068

    https://doi.org/10.1016/j.cam.2024.116068 [Citations: 0]

  19. Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

    Kang, Yuanyuan | Wang, Jindi | Yang, Yin

    Applied Numerical Mathematics, Vol. 206 (2024), Iss. P.190

    https://doi.org/10.1016/j.apnum.2024.08.005 [Citations: 0]

  20. A BDF2 energy‐stable scheme for the binary fluid‐surfactant hydrodynamic model

    Qin, Yuzhe | Chen, Rui | Zhang, Zhengru

    Mathematical Methods in the Applied Sciences, Vol. 45 (2022), Iss. 5 P.2776

    https://doi.org/10.1002/mma.7952 [Citations: 5]

  21. The stabilized-trigonometric scalar auxiliary variable approach for gradient flows and its efficient schemes

    Yang, Junxiang | Kim, Junseok

    Journal of Engineering Mathematics, Vol. 129 (2021), Iss. 1

    https://doi.org/10.1007/s10665-021-10155-x [Citations: 6]

  22. Structure-Preserving, Energy Stable Numerical Schemes for a Liquid Thin Film Coarsening Model

    Zhang, Juan | Wang, Cheng | Wise, Steven M. | Zhang, Zhengru

    SIAM Journal on Scientific Computing, Vol. 43 (2021), Iss. 2 P.A1248

    https://doi.org/10.1137/20M1375656 [Citations: 22]

  23. Energy stable and convergent BDF3-5 schemes for the molecular beam epitaxial model with slope selection

    Li, Juan | Sun, Hong | Zhao, Xuan

    International Journal of Computer Mathematics, Vol. 100 (2023), Iss. 7 P.1646

    https://doi.org/10.1080/00207160.2023.2208236 [Citations: 0]

  24. Phase-field modeling and linearly energy-stable Runge–Kutta algorithm of colloidal crystals on curved surfaces

    Yang, Junxiang | Li, Yibao | Kim, Junseok

    Journal of Computational and Applied Mathematics, Vol. 443 (2024), Iss. P.115750

    https://doi.org/10.1016/j.cam.2023.115750 [Citations: 1]

  25. Spectral deferred correction method for Landau–Brazovskii model with convex splitting technique

    Zhang, Donghang | Zhang, Lei

    Journal of Computational Physics, Vol. 491 (2023), Iss. P.112348

    https://doi.org/10.1016/j.jcp.2023.112348 [Citations: 1]

  26. A-Stable High-Order Block Implicit Methods for Parabolic Equations

    Li, Shishun | Wang, Jing-Yuan | Cai, Xiao-Chuan

    SIAM Journal on Numerical Analysis, Vol. 61 (2023), Iss. 4 P.1858

    https://doi.org/10.1137/22M152880X [Citations: 0]

  27. On novel linear schemes for the Cahn–Hilliard equation based on an improved invariant energy quadratization approach

    Chen, Rui | Gu, Shuting

    Journal of Computational and Applied Mathematics, Vol. 414 (2022), Iss. P.114405

    https://doi.org/10.1016/j.cam.2022.114405 [Citations: 5]

  28. Error Estimate of Exponential Time Differencing Runge-Kutta Scheme for the Epitaxial Growth Model Without Slope Selection

    Sun, Yabing | Zhou, Quan

    Journal of Scientific Computing, Vol. 93 (2022), Iss. 1

    https://doi.org/10.1007/s10915-022-01977-7 [Citations: 1]

  29. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

    Wang, Cheng

    Electronic Research Archive, Vol. 29 (2021), Iss. 5 P.2915

    https://doi.org/10.3934/era.2021019 [Citations: 1]

  30. An operator splitting scheme for numerical simulation of spinodal decomposition and microstructure evolution of binary alloys

    Shah, Abdullah | Ayub, Sana | Sohaib, Muhammad | Saeed, Sadia | Khan, Saher Akmal | Abbas, Suhail | Shah, Said Karim

    Heliyon, Vol. 9 (2023), Iss. 6 P.e16597

    https://doi.org/10.1016/j.heliyon.2023.e16597 [Citations: 1]

  31. An efficient nonstandard computer method to solve a compartmental epidemiological model for COVID-19 with vaccination and population migration

    Herrera-Serrano, Jorge E. | Macías-Díaz, Jorge E. | Medina-Ramírez, Iliana E. | Guerrero, J.A.

    Computer Methods and Programs in Biomedicine, Vol. 221 (2022), Iss. P.106920

    https://doi.org/10.1016/j.cmpb.2022.106920 [Citations: 2]

  32. Stability of high order finite difference and local discontinuous Galerkin schemes with explicit-implicit-null time-marching for high order dissipative and dispersive equations

    Tan, Meiqi | Cheng, Juan | Shu, Chi-Wang

    Journal of Computational Physics, Vol. 464 (2022), Iss. P.111314

    https://doi.org/10.1016/j.jcp.2022.111314 [Citations: 7]

  33. A fully discrete pseudospectral method for the nonlinear Fokker-Planck equations on the whole line

    Wang, Tian-jun | Chai, Guo

    Applied Numerical Mathematics, Vol. 174 (2022), Iss. P.17

    https://doi.org/10.1016/j.apnum.2022.01.003 [Citations: 5]

  34. An Explicit Adaptive Finite Difference Method for the Cahn–Hilliard Equation

    Ham, Seokjun | Li, Yibao | Jeong, Darae | Lee, Chaeyoung | Kwak, Soobin | Hwang, Youngjin | Kim, Junseok

    Journal of Nonlinear Science, Vol. 32 (2022), Iss. 6

    https://doi.org/10.1007/s00332-022-09844-3 [Citations: 4]

  35. A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches

    Li, Futuan | Yang, Jiang

    Computer Methods in Applied Mechanics and Engineering, Vol. 398 (2022), Iss. P.115195

    https://doi.org/10.1016/j.cma.2022.115195 [Citations: 2]

  36. An accurate and parallel method with post-processing boundedness control for solving the anisotropic phase-field dendritic crystal growth model

    Wang, Yan | Xiao, Xufeng | Feng, Xinlong

    Communications in Nonlinear Science and Numerical Simulation, Vol. 115 (2022), Iss. P.106717

    https://doi.org/10.1016/j.cnsns.2022.106717 [Citations: 13]

  37. Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows

    Chen, Yanping | Liu, Zhengguang | Meng, Xiaoqing

    Communications in Nonlinear Science and Numerical Simulation, Vol. 140 (2025), Iss. P.108379

    https://doi.org/10.1016/j.cnsns.2024.108379 [Citations: 0]

  38. Fully decoupled and energy stable BDF schemes for a class of Keller-Segel equations

    Wang, Shufen | Zhou, Simin | Shi, Shuxun | Chen, Wenbin

    Journal of Computational Physics, Vol. 449 (2022), Iss. P.110799

    https://doi.org/10.1016/j.jcp.2021.110799 [Citations: 5]

  39. An explicit conservative Saul’yev scheme for the Cahn–Hilliard equation

    Yang, Junxiang | Li, Yibao | Lee, Chaeyoung | Lee, Hyun Geun | Kwak, Soobin | Hwang, Youngjin | Xin, Xuan | Kim, Junseok

    International Journal of Mechanical Sciences, Vol. 217 (2022), Iss. P.106985

    https://doi.org/10.1016/j.ijmecsci.2021.106985 [Citations: 12]

  40. New third-order convex splitting methods and analysis for the phase field crystal equation

    Ye, Zhijian | Zheng, Zhoushun | Li, Zhilin

    Numerical Algorithms, Vol. (2024), Iss.

    https://doi.org/10.1007/s11075-024-01782-3 [Citations: 0]

  41. Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

    Zhang, Fan | Li, Dongfang | Sun, Hai-Wei

    Journal of Computational and Applied Mathematics, Vol. 427 (2023), Iss. P.115148

    https://doi.org/10.1016/j.cam.2023.115148 [Citations: 5]

  42. Energy Stability of BDF Methods up to Fifth-Order for the Molecular Beam Epitaxial Model Without Slope Selection

    Kang, Yuanyuan | Liao, Hong-lin

    Journal of Scientific Computing, Vol. 91 (2022), Iss. 2

    https://doi.org/10.1007/s10915-022-01830-x [Citations: 9]