March  2019, 24(3): 1341-1365. doi: 10.3934/dcdsb.2019019

On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

2. 

Xiamen University, School of Mathematical Sciences, Fujian Provincial Key Laboratory of Mathematical Modeling, and High Performance Scientific Computing, Xiamen, Fujian, China

3. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

4. 

Université de Picardie Jules Verne, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR CNRS 7352, Pôle Scientifique, 33, rue Saint Leu, F-80039 Amiens, France

* Corresponding author: Alain Miranville

Received  January 2018 Revised  March 2018 Published  January 2019

Our aim in this article is to study generalizations of the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global attractor. Furthermore, we give some numerical simulations, obtained with the $\mathtt{FreeFem++}$ software [24], comparing the nonconserved Caginalp phase-field model with regular and logarithmic nonlinear terms.

Citation: Ahmad Makki, Alain Miranville, Georges Sadaka. On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1341-1365. doi: 10.3934/dcdsb.2019019
References:
[1]

S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Eqns., 1 (2001), 69-84. doi: 10.1007/PL00001365. Google Scholar

[2]

S. AizicoviciE. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215. Google Scholar

[3]

D. BrochetX. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173. Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. Google Scholar

[5]

G. Caginalp, An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. Google Scholar

[7]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. Google Scholar

[8]

P. J. ChenM. E. Gurtin and W. O. Williams, A note on a non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970. doi: 10.1007/BF01602278. Google Scholar

[9]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112. doi: 10.1007/BF01591120. Google Scholar

[10]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129. Google Scholar

[11]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar

[12]

L. CherfilsS. Gatti and A. Miranville, A doubly nonlinear parabolic equation with a singular potential, Discrete Contin. Dyn. Systems S, 4 (2011), 51-66. doi: 10.3934/dcdss.2011.4.51. Google Scholar

[13]

R. ChillE Fasangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431. Google Scholar

[14]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving medis, Phys. Review Letters, 94 (2005), 154301.Google Scholar

[15]

B. Doumbé, Etude de modèles de champs de phase de type Caginalp, Université de Poitiers, 2013.Google Scholar

[16]

A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226. doi: 10.1080/01495739.2011.608313. Google Scholar

[17]

C. G. Gal and M. Grasselli, The nonisothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Systems A, 22 (2008), 1009-1040. doi: 10.3934/dcds.2008.22.1009. Google Scholar

[18]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in Differential Equations: Inverse and Direct Problems (Proceedings of the Workshop "Evolutiob Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004), A series of Lecture notes in pure and applied mathematics, 251, A. Favini and A. Lorenzi eds., Chapman & Hall, 2006,149–170. doi: 10.1201/9781420011135.ch9. Google Scholar

[19]

M. GrasselliA. MiranvilleV. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560. Google Scholar

[20]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems A, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67. Google Scholar

[21]

M. Grasselli and V. Pata, Existence of a universal attractor for a fully hyperbolic phase-field system, J. Evol. Eqns., 4 (2004), 27-51. doi: 10.1007/s00028-003-0074-2. Google Scholar

[22]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. Google Scholar

[23]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012. Google Scholar

[24]

F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, Freefem++ Manual, 2012.Google Scholar

[25]

J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169. doi: 10.1016/j.jmaa.2007.09.041. Google Scholar

[26]

J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182. doi: 10.1002/mma.1092. Google Scholar

[27]

A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Systems S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271. Google Scholar

[28]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061. Google Scholar

[29]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182. Google Scholar

[30]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar

[31]

A. Miranville and R. Quintanilla, A type $\rm III$ phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016. Google Scholar

[32]

A. Miranville and R. Quintanilla, A generalization of the Allen-Cahn equation, IMA J. Appl. Math., 80 (2015), 410-430. doi: 10.1093/imamat/hxt044. Google Scholar

[33]

A. Miranville and R. Quintanilla, A Caginalp phase-field system based on type Ⅲ heat conduction with two temperatures, Quart. Appl. Math., 74 (2016), 375-398. doi: 10.1090/qam/1430. Google Scholar

[34]

A. Miranville and R. Quintanilla, On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397. doi: 10.1002/mma.3867. Google Scholar

[35]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electronic J. Diff. Eqns., 2002 (2002), 1-28. Google Scholar

[36]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. Google Scholar

[37]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. Google Scholar

[38]

G. Sadaka, Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case, J. Numer. Math., 20 (2012), 303-324. doi: 10.1515/jnum-2012-0016. Google Scholar

[39]

H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390. doi: 10.1093/imamat/hxh101. Google Scholar

[40]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683. Google Scholar

show all references

References:
[1]

S. Aizicovici and E. Feireisl, Long-time stabilization of solutions to a phase-field model with memory, J. Evol. Eqns., 1 (2001), 69-84. doi: 10.1007/PL00001365. Google Scholar

[2]

S. AizicoviciE. Feireisl and F. Issard-Roch, Long-time convergence of solutions to a phase-field system, Math. Methods Appl. Sci., 24 (2001), 277-287. doi: 10.1002/mma.215. Google Scholar

[3]

D. BrochetX. Chen and D. Hilhorst, Finite dimensional exponential attractors for the phase-field model, Appl. Anal., 49 (1993), 197-212. doi: 10.1080/00036819108840173. Google Scholar

[4]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. Google Scholar

[5]

G. Caginalp, An analysis of a phase-field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. Google Scholar

[7]

P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Appl. Math. Phys. (ZAMP), 19 (1968), 614-627. Google Scholar

[8]

P. J. ChenM. E. Gurtin and W. O. Williams, A note on a non-simple heat conduction, J. Appl. Math. Phys. (ZAMP), 19 (1968), 969-970. doi: 10.1007/BF01602278. Google Scholar

[9]

P. J. ChenM. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Appl. Math. Phys. (ZAMP), 20 (1969), 107-112. doi: 10.1007/BF01591120. Google Scholar

[10]

L. Cherfils and A. Miranville, Some results on the asymptotic behavior of the Caginalp system with singular potentials, Adv. Math. Sci. Appl., 17 (2007), 107-129. Google Scholar

[11]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar

[12]

L. CherfilsS. Gatti and A. Miranville, A doubly nonlinear parabolic equation with a singular potential, Discrete Contin. Dyn. Systems S, 4 (2011), 51-66. doi: 10.3934/dcdss.2011.4.51. Google Scholar

[13]

R. ChillE Fasangovà and J. Prüss, Convergence to steady states of solutions of the Cahn-Hilliard equation with dynamic boundary conditions, Math. Nachr., 279 (2006), 1448-1462. doi: 10.1002/mana.200410431. Google Scholar

[14]

C. I. Christov and P. M. Jordan, Heat conduction paradox involving second-sound propagation in moving medis, Phys. Review Letters, 94 (2005), 154301.Google Scholar

[15]

B. Doumbé, Etude de modèles de champs de phase de type Caginalp, Université de Poitiers, 2013.Google Scholar

[16]

A. S. El-Karamany and M. A. Ezzat, On the two-temperature Green-Naghdi thermoelasticity theories, J. Thermal Stresses, 34 (2011), 1207-1226. doi: 10.1080/01495739.2011.608313. Google Scholar

[17]

C. G. Gal and M. Grasselli, The nonisothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Systems A, 22 (2008), 1009-1040. doi: 10.3934/dcds.2008.22.1009. Google Scholar

[18]

S. Gatti and A. Miranville, Asymptotic behavior of a phase-field system with dynamic boundary conditions, in Differential Equations: Inverse and Direct Problems (Proceedings of the Workshop "Evolutiob Equations: Inverse and Direct Problems", Cortona, June 21-25, 2004), A series of Lecture notes in pure and applied mathematics, 251, A. Favini and A. Lorenzi eds., Chapman & Hall, 2006,149–170. doi: 10.1201/9781420011135.ch9. Google Scholar

[19]

M. GrasselliA. MiranvilleV. Pata and S. Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280 (2007), 1475-1509. doi: 10.1002/mana.200510560. Google Scholar

[20]

M. GrasselliA. Miranville and G. Schimperna, The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials, Discrete Contin. Dyn. Systems A, 28 (2010), 67-98. doi: 10.3934/dcds.2010.28.67. Google Scholar

[21]

M. Grasselli and V. Pata, Existence of a universal attractor for a fully hyperbolic phase-field system, J. Evol. Eqns., 4 (2004), 27-51. doi: 10.1007/s00028-003-0074-2. Google Scholar

[22]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. Google Scholar

[23]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012. Google Scholar

[24]

F. Hecht, O. Pironneau, A. Le Hyaric and K. Ohtsuka, Freefem++ Manual, 2012.Google Scholar

[25]

J. Jiang, Convergence to equilibrium for a parabolic-hyperbolic phase-field model with Cattaneo heat flux law, J. Math. Anal. Appl., 341 (2008), 149-169. doi: 10.1016/j.jmaa.2007.09.041. Google Scholar

[26]

J. Jiang, Convergence to equilibrium for a fully hyperbolic phase field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32 (2009), 1156-1182. doi: 10.1002/mma.1092. Google Scholar

[27]

A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Systems S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271. Google Scholar

[28]

A. Miranville and R. Quintanilla, A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal. TMA, 71 (2009), 2278-2290. doi: 10.1016/j.na.2009.01.061. Google Scholar

[29]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182. Google Scholar

[30]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar

[31]

A. Miranville and R. Quintanilla, A type $\rm III$ phase-field system with a logarithmic potential, Appl. Math. Letters, 24 (2011), 1003-1008. doi: 10.1016/j.aml.2011.01.016. Google Scholar

[32]

A. Miranville and R. Quintanilla, A generalization of the Allen-Cahn equation, IMA J. Appl. Math., 80 (2015), 410-430. doi: 10.1093/imamat/hxt044. Google Scholar

[33]

A. Miranville and R. Quintanilla, A Caginalp phase-field system based on type Ⅲ heat conduction with two temperatures, Quart. Appl. Math., 74 (2016), 375-398. doi: 10.1090/qam/1430. Google Scholar

[34]

A. Miranville and R. Quintanilla, On the Caginalp phase-field systems with two temperatures and the Maxwell-Cattaneo law, Math. Methods Appl. Sci., 39 (2016), 4385-4397. doi: 10.1002/mma.3867. Google Scholar

[35]

A. Miranville and S. Zelik, Robust exponential attractors for singularly perturbed phase-field type equations, Electronic J. Diff. Eqns., 2002 (2002), 1-28. Google Scholar

[36]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735. doi: 10.1002/mma.590. Google Scholar

[37]

R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. Google Scholar

[38]

G. Sadaka, Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case, J. Numer. Math., 20 (2012), 303-324. doi: 10.1515/jnum-2012-0016. Google Scholar

[39]

H. M. Youssef, Theory of two-temperature-generalized thermoelasticity, IMA J. Appl. Math., 71 (2006), 383-390. doi: 10.1093/imamat/hxh101. Google Scholar

[40]

Z. Zhang, Asymptotic behavior of solutions to the phase-field equations with Neumann boundary conditions, Comm. Pure Appl. Anal., 4 (2005), 683-693. doi: 10.3934/cpaa.2005.4.683. Google Scholar

Figure 1.  Solution $ u $ with $ f_H = .1 $ and logarithmic potential
Figure 2.  Solution $u$ with $f_H = .1$ and cubic potential
Figure 3.  Solution $u$ with $f_H = .1$ and logarithmic potential
Figure 4.  Solution $u$ with $f_H = .1$ and cubic potential
Figure 5.  Solution $u$ with $f_H = 1.5$ and logarithmic potential
Figure 6.  Solution $u$ with $f_H = 1.5$ and cubic potential
Figure 7.  Solution $H$ with $f_H = 1.5$ and logarithmic potential
Figure 8.  Solution $H$ with $f_H = 1.5$ and cubic potential
Table 1.  $ L^2 $, $ H^1 $ norm and error for $ u $ and $ H $ for the nonconserved Caginalp phase-field system with $ \varepsilon = .1,\ \tau = .03 $ and $ f(s) = .83(s^3-.5^2s) $
$ 10^{2}\cdot\delta t $ CPU time $ N_{L^2}(u) $ rate $ N_{L^2}(H) $ rate $ N_{H^1}(u) $ rate $ N_{H^1}(H) $ rate
1/1 00:00:02 0.00147 - 0.00124 - 0.0573 - 0.05544 -
1/4 00:00:22 0.00038 0.98 0.0003 0.98 0.02927 0.48 0.02828 0.49
1/16 00:05:58 9.5e-05 0.99 8.1e-05 0.99 0.01472 0.49 0.01422 0.49
1/64 01:50:22 2.4e-05 0.99 2e-05 0.99 0.00737 0.49 0.00711 0.49
1/256 22:06:19 6e-06 1 5e-06 0.99 0.00369 0.5 0.00357 0.49
$ 10^{2}\cdot\delta t $ CPU time $ N_{L^2}(u) $ rate $ N_{L^2}(H) $ rate $ N_{H^1}(u) $ rate $ N_{H^1}(H) $ rate
1/1 00:00:02 0.00147 - 0.00124 - 0.0573 - 0.05544 -
1/4 00:00:22 0.00038 0.98 0.0003 0.98 0.02927 0.48 0.02828 0.49
1/16 00:05:58 9.5e-05 0.99 8.1e-05 0.99 0.01472 0.49 0.01422 0.49
1/64 01:50:22 2.4e-05 0.99 2e-05 0.99 0.00737 0.49 0.00711 0.49
1/256 22:06:19 6e-06 1 5e-06 0.99 0.00369 0.5 0.00357 0.49
Table 2.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .1, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:020.00146-0.00124-0.05725-0.05544-
1/400:00:250.000380.980.00030.980.029270.480.028280.49
1/1600:07:449.5e-050.998.1e-050.990.014720.490.014220.49
1/6401:52:092.4e-050.992e-050.990.007370.490.007110.49
1/25623:06:556e-0615e-060.990.003690.50.003570.49
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:020.00146-0.00124-0.05725-0.05544-
1/400:00:250.000380.980.00030.980.029270.480.028280.49
1/1600:07:449.5e-050.998.1e-050.990.014720.490.014220.49
1/6401:52:092.4e-050.992e-050.990.007370.490.007110.49
1/25623:06:556e-0615e-060.990.003690.50.003570.49
Table 3.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .01, \ \tau = .03$ and $f(s) = .83(s^3-.5^2s)$
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00042-0.00039-0.02458-0.02397-
1/400:00:220.000110.980.00010.980.012490.490.012150.49
1/1600:07:092.7e-050.992.5e-050.990.006270.490.006110.49
1/6401:27:207e-060.996e-060.990.003140.490.003060.49
1/25622:16:472e-0612e-060.990.001570.50.001540.49
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00042-0.00039-0.02458-0.02397-
1/400:00:220.000110.980.00010.980.012490.490.012150.49
1/1600:07:092.7e-050.992.5e-050.990.006270.490.006110.49
1/6401:27:207e-060.996e-060.990.003140.490.003060.49
1/25622:16:472e-0612e-060.990.001570.50.001540.49
Table 4.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .01, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00042-0.00039-0.02457-0.02397-
1/400:00:240.000110.980.00010.980.012480.490.012150.49
1/1600:07:412.7e-050.992.5e-050.990.006270.490.006110.49
1/6401:26:117e-060.996e-060.990.003140.490.003060.49
1/25623:06:302e-0612e-060.990.001570.50.001540.49
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00042-0.00039-0.02457-0.02397-
1/400:00:240.000110.980.00010.980.012480.490.012150.49
1/1600:07:412.7e-050.992.5e-050.990.006270.490.006110.49
1/6401:26:117e-060.996e-060.990.003140.490.003060.49
1/25623:06:302e-0612e-060.990.001570.50.001540.49
Table 5.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .001, \ \tau = .03$ and $f(s) = .83(s^3-.5^2s)$.
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00013-0.00012-0.01227-0.01186-
1/400:00:223.4e-050.983.2e-050.980.006250.490.006030.49
1/1600:07:199e-060.998e-060.990.003140.490.003030.49
1/6401:29:472e-060.992e-060.990.001570.490.001520.49
1/25622:23:291e-0611e-060.990.000790.50.000760.49
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00013-0.00012-0.01227-0.01186-
1/400:00:223.4e-050.983.2e-050.980.006250.490.006030.49
1/1600:07:199e-060.998e-060.990.003140.490.003030.49
1/6401:29:472e-060.992e-060.990.001570.490.001520.49
1/25622:23:291e-0611e-060.990.000790.50.000760.49
Table 6.  $L^2$, $H^1$ norm and error for $u$ and $H$ for the nonconserved Caginalp phase-field system with $\varepsilon = .001, \ \tau = .03$ and $f(s) = -2 \kappa_0 s+\kappa_1 \ln\left(\dfrac{1+s}{1-s}\right)$.
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00013-0.00012-0.01226-0.011863-
1/400:00:243.4e-050.983.2e-050.980.006240.490.006030.49
1/1600:07:449e-060.998e-060.990.003140.490.003030.49
1/6401:29:122e-060.992e-060.990.001570.490.001520.49
1/25623:22:361e-0611e-060.990.000790.50.000760.49
$10^{2}\cdot\delta t$CPU time$N_{L^2}(u)$rate$N_{L^2}(H)$rate$N_{H^1}(u)$rate$N_{H^1}(H)$rate
1/100:00:010.00013-0.00012-0.01226-0.011863-
1/400:00:243.4e-050.983.2e-050.980.006240.490.006030.49
1/1600:07:449e-060.998e-060.990.003140.490.003030.49
1/6401:29:122e-060.992e-060.990.001570.490.001520.49
1/25623:22:361e-0611e-060.990.000790.50.000760.49
Table 7.  Comparison of the convergence of the solution $u$ with $\varepsilon = .1$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
logexplose$-.41$$[-.37, .37]$$.41$.44.62$.94$explose
polexplose$-.40$$[-.37, .37]$$.40$.43.64exploseexplose
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
logexplose$-.41$$[-.37, .37]$$.41$.44.62$.94$explose
polexplose$-.40$$[-.37, .37]$$.40$.43.64exploseexplose
Table 8.  Comparison of the convergence of the solution $u$ with $\varepsilon = .01$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log$-.76$$[-.49, .49]$$[-.49, .49]$$[-.49, .49]$.49.52$.67$$.76$
pol$-.86$$[-.49, .49]$$[-.49, .49]$$[-.49, .49]$$.49$$.52$$.70$$.86$
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log$-.76$$[-.49, .49]$$[-.49, .49]$$[-.49, .49]$.49.52$.67$$.76$
pol$-.86$$[-.49, .49]$$[-.49, .49]$$[-.49, .49]$$.49$$.52$$.70$$.86$
Table 9.  Comparison of the convergence of the solution $u$ with $\varepsilon = .001$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log$-.56$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$.53$$.56$
pol$-.57$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$.53$$.57$
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log$-.56$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$.53$$.56$
pol$-.57$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$[-.50, .50]$$.53$$.57$
Table 10.  Comparison of the convergence of the solution $u$ with $\varepsilon = .1$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
logexploseexploseexploseexplose.92.94.99explose
polexploseexploseexploseexploseexplose.94exploseexplose
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
logexploseexploseexploseexplose.92.94.99explose
polexploseexploseexploseexploseexplose.94exploseexplose
Table 11.  comp_log_eps_0p01_ln6} Comparison of the convergence of the solution $u$ with $\varepsilon = .01$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log-.95$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$explose.94.95
polexploseexploseexploseexploseexploseexplose.96explose
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log-.95$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$explose.94.95
polexploseexploseexploseexploseexploseexplose.96explose
Table 12.  Comparison of the convergence of the solution $u$ with $\varepsilon = .001$.
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log-.93$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$explose.93
pol-.94exploseexploseexploseexploseexploseexplose.94
$f_H$$-35$$-.1$$0$$.1$$.2$$1.5$$15$$35$
log-.93$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$$[-.93, .93]$explose.93
pol-.94exploseexploseexploseexploseexploseexplose.94
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