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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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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