doi: 10.3934/dcdsb.2021246
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Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

Department of Mathematics, The George Washington University, Washington, D.C. 20052, USA

*Corresponding author: Yanxiang Zhao (yxzhao@gwu.edu)

Received  March 2021 Revised  August 2021 Early access October 2021

In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.

Citation: Hyunjung Choi, Yanxiang Zhao. Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021246
References:
[1]

F. S. Bates and G. H. Fredrickson, Block copolymer thermodynamics: Theory and experiment, Annu. Rev. Phys. Chem., 41 (1990), 525-557.  doi: 10.1146/annurev.pc.41.100190.002521.

[2]

F. S. Bates and G. H. Fredrickson, Block copolymers–designer soft materials, Phys. Today, 52 (1999), 32pp. doi: 10.1063/1.882522.

[3]

B. BenesovaC. Melcher and E. Suli, An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations, SIAM J. Numer. Anal., 52 (2014), 1466-1496.  doi: 10.1137/130940736.

[4]

L. Caffarelli and N. E. Muler, An ${L}^{\infty}$ bound for solutions of the Cahn-Hilliard equation, Arch. Rational. Mech. Anal., 133 (1995), 129-144.  doi: 10.1007/BF00376814.

[5]

B. Camley, Y. Zhao, B. Li, H. Levine and W. -J. Rappel, Periodic migration in a physical model of cells on micropatterns, Phys. Rev. Lett., 111 (2013). doi: 10.1103/PhysRevLett. 111.158102.

[6]

W. ChenS. CondeC. WangX. Wang and S. 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.

[7]

C. Chen, X. Li, J. Zhang and X. Yang, Efficient linear, decoupled, and unconditionally stable scheme for a ternary Cahn-Hilliard type Nakazawa-Ohta phase-field model for tri-block copolymers, Appl. Math. Comput., 388 (2021), 19pp. doi: 10.1016/j. amc. 2020.125463.

[8]

Q. ChengX. Yang and J. Shen, Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model, J. Comput. Phys., 341 (2017), 44-60.  doi: 10.1016/j.jcp.2017.04.010.

[9]

L. Dong, C. Wang, S. Wise and Z. Zhang, A positivity preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters, J. Comput. Phys., 442 (2021), 29pp. doi: 10.1016/j. jcp. 2021.110451.

[10]

Q. DuL. JuX. Li and Z. Qiao, Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation, J. Comput. Phys., 363 (2018), 39-54.  doi: 10.1016/j.jcp.2018.02.023.

[11]

Q. DuL. JuX. Li and Z. Qiao, Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes, SIAM Review, 63 (2021), 317-359.  doi: 10.1137/19M1243750.

[12]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and Mathematical Models of Microstructural Evolution, Mater. Res. Soc. Sympos. Proc., 529 (1998), 39-46.  doi: 10.1557/PROC-529-39.

[13]

X. FengT. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80.  doi: 10.4208/eajam.200113.220213a.

[14]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.

[15]

Z. GuanJ. LowengrubC. Wang and S. 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.

[16]

Z. GuanC. Wang and S. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406.  doi: 10.1007/s00211-014-0608-2.

[17]

Z. GuanJ. Lowengrub and C. Wang, Convergence analysis for second-order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations, Math. Meth. Appl. Sci., 40 (2017), 6836-6863.  doi: 10.1002/mma.4497.

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E. Helfand and Z. R. Wasserman, Block copolymer theory. 4. narrow interphase approximation, Macromolecules, 9 (1976), 879-888. 

[19]

Z. HuS. WiseC. Wang and J. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid scheme for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020.

[20]

L. JuX. LiZ. Qiao and H. Zhang, Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comput., 87 (2018), 1859-1885.  doi: 10.1090/mcom/3262.

[21]

L. Leibler, Theory of microphase separation in block copolymers, Macromolecules, 13 (1980), 1602-1617.  doi: 10.1021/ma60078a047.

[22]

X. LiZ. Qiao and C. Wang, Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, Math. Comput., 90 (2021), 171-188.  doi: 10.1090/mcom/3578.

[23]

J. Li, L. Ju, Y. Cai and X. Feng, Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint, J. Sci. Comput., 87 (2021), 32pp. doi: 10.1007/s10915-021-01512-0.

[24]

D. J. Meier, Theory of block copolymers. I. Domain formation in A-B block copolymers, J. Polym. Sci., Part C: Polym. Symp., 26 (1969), 81-98. 

[25]

X. MengZ. QiaoC. Wang and Z. Zhang, Artificial regularization parameter analysis for the no-slope-selection epitaxial thin film model, CSIAM Tran. App. Math., 1 (2020), 441-462.  doi: 10.4208/csiam-am.2020-0015.

[26]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.

[27]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Dis. Cont. Dyn. Syst. A, 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

[28]

J. Shen, T. Tang and L. -L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[29]

J. ShenC. WangX. Wang and S. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.  doi: 10.1137/110822839.

[30]

J. ShenJ. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61 (2019), 474-506.  doi: 10.1137/17M1150153.

[31]

C. Wang and S. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.  doi: 10.1137/090752675.

[32]

C. WangX. Ren and Y. Zhao, Bubble assemblies in termary systems with long range interaction, Commun. Math. Sci., 17 (2019), 2309-2324.  doi: 10.4310/CMS.2019.v17.n8.a10.

[33]

C. Wang, Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations, Elec. Res. Arch., (2020).

[34]

S. WiseC. Wang and J. 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.

[35]

Z. Xia and X. Yang, A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation, Dis. Cont. Dyn. Sys. B, 25 (2020), 3749-3763.  doi: 10.3934/dcdsb.2020089.

[36]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal, 44 (2006), 1759-1779.  doi: 10.1137/050628143.

[37]

X. Xu and Y. Zhao, Energy stable semi-implicit schemes for Allen-Cahn-Ohta-Kawasaki model in binary system, J. Sci. Comput., 80 (2019), 1656-1680.  doi: 10.1007/s10915-019-00993-4.

[38]

X. Xu and Y. Zhao, Maximum principle preserving schemes for binary systems with long-range interactions, J. Sci. Comput., 84 (2020), 34pp. doi: 10.1007/s10915-020-01286-x.

[39]

X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316.  doi: 10.1016/j.jcp.2016.09.029.

[40]

J. Zhang, C. Chen, X. Yang and K. Pan, Efficient numerical scheme for a penalized Allen-Cahn type Ohta-Kawasaki phase-field model for diblock copolymers, J. Comput. Appl. Math., 378 (2020), 23pp. doi: 10.1016/j. cam. 2020.112905.

[41]

Y. ZhaoY. MaH. SunB. Li and Q. Du, A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation, Commun. Math. Sci., 16 (2018), 1203-1223.  doi: 10.4310/CMS.2018.v16.n5.a2.

show all references

References:
[1]

F. S. Bates and G. H. Fredrickson, Block copolymer thermodynamics: Theory and experiment, Annu. Rev. Phys. Chem., 41 (1990), 525-557.  doi: 10.1146/annurev.pc.41.100190.002521.

[2]

F. S. Bates and G. H. Fredrickson, Block copolymers–designer soft materials, Phys. Today, 52 (1999), 32pp. doi: 10.1063/1.882522.

[3]

B. BenesovaC. Melcher and E. Suli, An implicit midpoint spectral approximation of nonlocal Cahn-Hilliard equations, SIAM J. Numer. Anal., 52 (2014), 1466-1496.  doi: 10.1137/130940736.

[4]

L. Caffarelli and N. E. Muler, An ${L}^{\infty}$ bound for solutions of the Cahn-Hilliard equation, Arch. Rational. Mech. Anal., 133 (1995), 129-144.  doi: 10.1007/BF00376814.

[5]

B. Camley, Y. Zhao, B. Li, H. Levine and W. -J. Rappel, Periodic migration in a physical model of cells on micropatterns, Phys. Rev. Lett., 111 (2013). doi: 10.1103/PhysRevLett. 111.158102.

[6]

W. ChenS. CondeC. WangX. Wang and S. 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.

[7]

C. Chen, X. Li, J. Zhang and X. Yang, Efficient linear, decoupled, and unconditionally stable scheme for a ternary Cahn-Hilliard type Nakazawa-Ohta phase-field model for tri-block copolymers, Appl. Math. Comput., 388 (2021), 19pp. doi: 10.1016/j. amc. 2020.125463.

[8]

Q. ChengX. Yang and J. Shen, Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model, J. Comput. Phys., 341 (2017), 44-60.  doi: 10.1016/j.jcp.2017.04.010.

[9]

L. Dong, C. Wang, S. Wise and Z. Zhang, A positivity preserving, energy stable scheme for a ternary Cahn-Hilliard system with the singular interfacial parameters, J. Comput. Phys., 442 (2021), 29pp. doi: 10.1016/j. jcp. 2021.110451.

[10]

Q. DuL. JuX. Li and Z. Qiao, Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation, J. Comput. Phys., 363 (2018), 39-54.  doi: 10.1016/j.jcp.2018.02.023.

[11]

Q. DuL. JuX. Li and Z. Qiao, Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes, SIAM Review, 63 (2021), 317-359.  doi: 10.1137/19M1243750.

[12]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and Mathematical Models of Microstructural Evolution, Mater. Res. Soc. Sympos. Proc., 529 (1998), 39-46.  doi: 10.1557/PROC-529-39.

[13]

X. FengT. Tang and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80.  doi: 10.4208/eajam.200113.220213a.

[14]

S. Gottlieb and C. Wang, Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers' equation, J. Sci. Comput., 53 (2012), 102-128.  doi: 10.1007/s10915-012-9621-8.

[15]

Z. GuanJ. LowengrubC. Wang and S. 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.

[16]

Z. GuanC. Wang and S. Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math., 128 (2014), 377-406.  doi: 10.1007/s00211-014-0608-2.

[17]

Z. GuanJ. Lowengrub and C. Wang, Convergence analysis for second-order accurate schemes for the periodic nonlocal Allen-Cahn and Cahn-Hilliard equations, Math. Meth. Appl. Sci., 40 (2017), 6836-6863.  doi: 10.1002/mma.4497.

[18]

E. Helfand and Z. R. Wasserman, Block copolymer theory. 4. narrow interphase approximation, Macromolecules, 9 (1976), 879-888. 

[19]

Z. HuS. WiseC. Wang and J. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid scheme for the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.  doi: 10.1016/j.jcp.2009.04.020.

[20]

L. JuX. LiZ. Qiao and H. Zhang, Energy stability and error estimates of exponential time differencing schemes for the epitaxial growth model without slope selection, Math. Comput., 87 (2018), 1859-1885.  doi: 10.1090/mcom/3262.

[21]

L. Leibler, Theory of microphase separation in block copolymers, Macromolecules, 13 (1980), 1602-1617.  doi: 10.1021/ma60078a047.

[22]

X. LiZ. Qiao and C. Wang, Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn-Hilliard equation, Math. Comput., 90 (2021), 171-188.  doi: 10.1090/mcom/3578.

[23]

J. Li, L. Ju, Y. Cai and X. Feng, Unconditionally maximum bound principle preserving linear schemes for the conservative Allen-Cahn equation with nonlocal constraint, J. Sci. Comput., 87 (2021), 32pp. doi: 10.1007/s10915-021-01512-0.

[24]

D. J. Meier, Theory of block copolymers. I. Domain formation in A-B block copolymers, J. Polym. Sci., Part C: Polym. Symp., 26 (1969), 81-98. 

[25]

X. MengZ. QiaoC. Wang and Z. Zhang, Artificial regularization parameter analysis for the no-slope-selection epitaxial thin film model, CSIAM Tran. App. Math., 1 (2020), 441-462.  doi: 10.4208/csiam-am.2020-0015.

[26]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.

[27]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Dis. Cont. Dyn. Syst. A, 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

[28]

J. Shen, T. Tang and L. -L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[29]

J. ShenC. WangX. Wang and S. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.  doi: 10.1137/110822839.

[30]

J. ShenJ. Xu and J. Yang, A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61 (2019), 474-506.  doi: 10.1137/17M1150153.

[31]

C. Wang and S. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49 (2011), 945-969.  doi: 10.1137/090752675.

[32]

C. WangX. Ren and Y. Zhao, Bubble assemblies in termary systems with long range interaction, Commun. Math. Sci., 17 (2019), 2309-2324.  doi: 10.4310/CMS.2019.v17.n8.a10.

[33]

C. Wang, Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations, Elec. Res. Arch., (2020).

[34]

S. WiseC. Wang and J. 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.

[35]

Z. Xia and X. Yang, A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation, Dis. Cont. Dyn. Sys. B, 25 (2020), 3749-3763.  doi: 10.3934/dcdsb.2020089.

[36]

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal, 44 (2006), 1759-1779.  doi: 10.1137/050628143.

[37]

X. Xu and Y. Zhao, Energy stable semi-implicit schemes for Allen-Cahn-Ohta-Kawasaki model in binary system, J. Sci. Comput., 80 (2019), 1656-1680.  doi: 10.1007/s10915-019-00993-4.

[38]

X. Xu and Y. Zhao, Maximum principle preserving schemes for binary systems with long-range interactions, J. Sci. Comput., 84 (2020), 34pp. doi: 10.1007/s10915-020-01286-x.

[39]

X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316.  doi: 10.1016/j.jcp.2016.09.029.

[40]

J. Zhang, C. Chen, X. Yang and K. Pan, Efficient numerical scheme for a penalized Allen-Cahn type Ohta-Kawasaki phase-field model for diblock copolymers, J. Comput. Appl. Math., 378 (2020), 23pp. doi: 10.1016/j. cam. 2020.112905.

[41]

Y. ZhaoY. MaH. SunB. Li and Q. Du, A new phase-field approach to variational implicit solvation of charged molecules with the Coulomb-field approximation, Commun. Math. Sci., 16 (2018), 1203-1223.  doi: 10.4310/CMS.2018.v16.n5.a2.

Figure 1.  (Color online) Coarsening dynamics in binary system with $ \gamma = 5000 $ (top) and $ \gamma = 8000 $ (bottom). The larger the repulsive force $ \gamma $ is, the more bubbles are generated from random initial
Figure 2.  (Color online) Coarsening dynamics in ternary system with various $ \gamma_{11} = \gamma_{22} $ and $ \gamma_{12} = \gamma_{21} $: $ \gamma_{11} = \gamma_{22} = 2000 $ and $ \gamma_{12} = \gamma_{21} = 0 $ (top), $ \gamma_{11} = \gamma_{22} = 2000 $, and $ \gamma_{12} = \gamma_{21} = 1350 $ (middle), $ \gamma_{11} = \gamma_{22} = 10000 $, and $ \gamma_{12} = \gamma_{21} = 12000 $ (bottom). When $ \gamma_{12} = \gamma_{21} $ is small, a double-bubble hexagonal pattern is formed, with the same polarity direction. As $ \gamma_{12} = \gamma_{21} $ becomes large, double-bubbles are broken, and a perfectly mixed red/yellow single-bubble square pattern is formed. Even larger $ \gamma_{12} = \gamma_{21} $ will further push away the two colors, leading to a complete separation between red and yellow, each of which has its own hexagonal pattern
Table 1.  Numerical rates of convergence in binary system (pACOK) with parameters $ \omega = 0.1 $, $ M = 1000 $, $ \gamma = 100 $, $ \kappa = 500,000 $, and $ \beta = 500 $
$ \epsilon = 20h $ $ \epsilon = 10h $ $ \epsilon = 5h $
$ \tau $ Error Rate Error Rate Error Rate
6.25000e-4 1.7780e-1 2.7611e-1 3.0359e-1
3.12500e-4 5.7956e-2 1.6173 1.3601e-1 1.0215 2.0887e-1 0.5395
1.56250e-4 1.5596e-2 1.8971 4.3307e-2 1.6510 9.0438e-1 1.2076
7.81250e-5 4.2290e-3 1.8794 1.1944e-2 1.8584 2.7308e-2 1.7276
3.90625e-5 1.2178e-3 1.7960 3.2868e-3 1.8615 7.4445e-3 1.8751
1e-6
(benchmark)
$ \epsilon = 20h $ $ \epsilon = 10h $ $ \epsilon = 5h $
$ \tau $ Error Rate Error Rate Error Rate
6.25000e-4 1.7780e-1 2.7611e-1 3.0359e-1
3.12500e-4 5.7956e-2 1.6173 1.3601e-1 1.0215 2.0887e-1 0.5395
1.56250e-4 1.5596e-2 1.8971 4.3307e-2 1.6510 9.0438e-1 1.2076
7.81250e-5 4.2290e-3 1.8794 1.1944e-2 1.8584 2.7308e-2 1.7276
3.90625e-5 1.2178e-3 1.7960 3.2868e-3 1.8615 7.4445e-3 1.8751
1e-6
(benchmark)
Table 2.  Numerical rates of convergence in ternary system (pACON) with parameters $ \omega_1 = \omega_2 = 0.1 $, $ M_1 = M_2 = 1000 $, $ \gamma_{11} = \gamma_{22} = 2000 $, $ \gamma_{12} = \gamma_{21} = 0 $, $ \kappa_1 = \kappa_2 = 1000 $, and $ \beta_1 = \beta_2 = 0 $
$ \epsilon = 20h $ $ \epsilon = 10h $ $ \epsilon = 5h $
$ \tau $ Error Rate Error Rate Error Rate
1.9531e-4 6.4752e-2 8.6449e-2 1.1016e-1
9.7656e-5 3.8307e-2 0.7573 5.2877e-2 0.7092 7.4234e-2 0.5694
4.8828e-5 2.0243e-2 0.9202 2.7753e-2 0.9300 4.0591e-2 0.8709
2.4414e-5 9.7032e-3 1.0609 1.3121e-2 1.0807 1.8854e-2 1.1063
1.2207e-5 4.0208e-3 1.2710 5.3810e-3 1.2860 7.6155e-3 1.3078
6.1035e-6 1.0766e-3 1.9010 1.4069e-3 1.9354 1.9639e-3 1.9552
1e-7
(benchmark)
$ \epsilon = 20h $ $ \epsilon = 10h $ $ \epsilon = 5h $
$ \tau $ Error Rate Error Rate Error Rate
1.9531e-4 6.4752e-2 8.6449e-2 1.1016e-1
9.7656e-5 3.8307e-2 0.7573 5.2877e-2 0.7092 7.4234e-2 0.5694
4.8828e-5 2.0243e-2 0.9202 2.7753e-2 0.9300 4.0591e-2 0.8709
2.4414e-5 9.7032e-3 1.0609 1.3121e-2 1.0807 1.8854e-2 1.1063
1.2207e-5 4.0208e-3 1.2710 5.3810e-3 1.2860 7.6155e-3 1.3078
6.1035e-6 1.0766e-3 1.9010 1.4069e-3 1.9354 1.9639e-3 1.9552
1e-7
(benchmark)
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