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On the $L^p$ regularity of solutions to the generalized Hunter-Saxton system

Efficient numerical schemes for two-dimensional Ginzburg-Landau equation in superconductivity

 1 School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China 2 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

* Corresponding author: Linghua Kong

Received  September 2018 Revised  January 2019 Published  July 2019

Fund Project: This work is supported by the NNSFC(Nos. 11271171, 11301234, and 11571181), the Natural Science Foundation of Jiangxi Province (Nos. 20161ACB20006, 20142BCB23009, 20181BAB201008), the State Key Laboratory of Scientific and Engineering Computing, CAS

The objective of this paper is to propose some high-order compact schemes for two-dimensional Ginzburg-Landau equation. The space is approximated by high-order compact methods to improve the computational efficiency. Based on Crank-Nicolson method in time, several temporal approximations are used starting from different viewpoints. The numerical characters of the new schemes, such as the existence and uniqueness, stability, convergence are investigated. Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works. In the numerical experiments, the role of some parameters in the model is considered and tested.

Citation: Linghua Kong, Liqun Kuang, Tingchun Wang. Efficient numerical schemes for two-dimensional Ginzburg-Landau equation in superconductivity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019141
References:
 [1] F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math., 17 (1965), 24-49. Google Scholar [2] C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B., 36 (1995), 313-324. doi: 10.1017/S0334270000010468. Google Scholar [3] Z. Cao, B. Guo and B. Wang, Global existence theory for the two-dimensional derivative Ginzburg-Landau equation, Chin. Sci. Bull., 43 (1998), 393-395. doi: 10.1007/BF02883716. Google Scholar [4] Z. Chen, Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity, Numer. Math., 76 (1997), 323-353. doi: 10.1007/s002110050266. Google Scholar [5] Z. Chen and S. Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity, SIAM J. Numer. Anal., 38 (2001), 1961-1985. doi: 10.1137/S0036142998349102. Google Scholar [6] C. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1998), 279-309. doi: 10.1088/0951-7715/1/2/001. Google Scholar [7] Q. Du, M. D. Gunzburger and J. S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34 (1992), 54-81. doi: 10.1137/1034003. Google Scholar [8] Q. Du, Finite element methods for the time-dependent Ginzberg-Landau model of superconductivity, Comput. Math. Appl., 27 (1994), 119-133. doi: 10.1016/0898-1221(94)90091-4. Google Scholar [9] Q. Du, Discrete gauge invariant approximations of a time dependent Ginzberg-Landau model of superconductivity, Math. Comput., 67 (1998), 965-986. doi: 10.1090/S0025-5718-98-00954-5. Google Scholar [10] L. C. Evans, Partial Differential Equations, Providence, American Mathematical Society, Rhode Island, 1998. doi: 10.1090/gsm/019. Google Scholar [11] H. Gao, B. Li and W. Sun, Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity, SIAM J. Numer. Anal., 52 (2014), 1183-1202. doi: 10.1137/130918678. Google Scholar [12] H. Gao and J. Duan, Asymptotic for the generalized two dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 247 (2000), 198-216. doi: 10.1006/jmaa.2000.6848. Google Scholar [13] J. M. Gnidaglia and B. Heron, Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28 (1987), 282-304. doi: 10.1016/0167-2789(87)90020-0. Google Scholar [14] L. Kong, J Hong, L. Ji and P. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Eq., 31 (2015), 1814-1843. doi: 10.1002/num.21969. Google Scholar [15] A. Y. Kolesov and N. K. Rosov, Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain, Theor. Math. Phys., 125 (2000), 1476-1488. doi: 10.1007/BF02551008. Google Scholar [16] L. Kuang, L. Kong, L. Wang and X. Zheng, The splitting high-order compact scheme for two-dimensional Ginzburg-Landau equation, J Jianxi Normal Univer., 40 (2017), 35-38. Google Scholar [17] S. K. Lele, Compact finite difference schemes with spectral-like solution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R. Google Scholar [18] C. D. Levemore and M. Oliver, The complex Ginzburg-Landau equation as a model problem, Lect. Appl. Math., 31 (1996), 141-190. Google Scholar [19] J. Li, High order compact ADI methods for the parabolic equations, Comput. Math. Appl., 52 (2006), 1343-1356. doi: 10.1016/j.camwa.2006.11.010. Google Scholar [20] Y. Li and B. Guo, Global existence of solutions to the derivative Ginzburg-Landau equation, J. Math. Anal. Appl., 249 (2000), 412-432. doi: 10.1006/jmaa.2000.6880. Google Scholar [21] B. Li, Convergence of a decoupled mixed FEM for the dynamic Ginzberg-Landau equations in nonsmooth domains with incompatible initial, Calclo, 54 (2017), 1441-1480. doi: 10.1007/s10092-017-0237-0. Google Scholar [22] B. Li and Z. Zhang, Mathematical and numerical analysis of time-dependent Ginzburg-Landau equations in nonconvex polygons based on hodge decomposition, Math. Comput., 86 (2017), 1579-1608. doi: 10.1090/mcom/3177. Google Scholar [23] S. Lü and Q. Lu, Exponential attractor for the 3D Ginzburg-Landau type equation, Nonlin. Anal., 67 (2007), 3116-3135. doi: 10.1016/j.na.2006.10.005. Google Scholar [24] S. Lü, Q. Lu and E. H. Twizell, Fourier spectral approximation to long-time behavior of the derivative three-dimensional Ginzburg-Landau equation, J. Comput. Appl. Math., 198 (2007), 167-186. doi: 10.1016/j.cam.2005.11.028. Google Scholar [25] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D., 41 (1990), 232-252. doi: 10.1016/0167-2789(90)90125-9. Google Scholar [26] Y. V. S. S. Sanyasiraju and N. Mishra, Spectral resolution exponential compact higher order scheme for convection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4737-4744. doi: 10.1016/j.cma.2008.06.013. Google Scholar [27] A Shokri and F Afshari, High-order compact ADI method using predictor-corrector scheme for 2D complex Ginzburg-Landau equation, Comput. Phys. Commun., 197 (2015), 43-50. doi: 10.1016/j.cpc.2015.08.005. Google Scholar [28] B. Wang, Existence of time periodic solutions for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 232 (1999), 394-412. doi: 10.1006/jmaa.1999.6283. Google Scholar [29] H. Wang, An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations, Comput. Phys. Comms., 181 (2010), 325-340. doi: 10.1016/j.cpc.2009.10.007. Google Scholar [30] L. Wang, Y. Zhou and L. Fu, The compact and modified ADI scheme for Schrödinger equations, J. Jiangxi Normal Univer., 40 (2016), 515-519. Google Scholar [31] T. Wang and B. Guo, Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation, Numer. Meth. Part. Diff. Eq., 27 (2011), 1340-1363. doi: 10.1002/num.20588. Google Scholar [32] P. Wang and C. Huang, An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation, BIT Numer. Math., 58 (2018), 783-805. doi: 10.1007/s10543-018-0698-9. Google Scholar [33] X. Wang, W. Li and L. Mao, On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation $AX+XB = C$., Comput. Math. with Appl., 66 (2013), 2352-2361. doi: 10.1016/j.camwa.2013.09.011. Google Scholar [34] Q. Xu and Q. Chang, Difference methods for computing the Ginzburg-Landau equation in two dimensions, Numer. Meth. Part. Diff. Eq., 27 (2011), 507-528. doi: 10.1002/num.20535. Google Scholar [35] M. Zhan, Existence of periodic solutions for Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 249 (2000), 614-625. doi: 10.1006/jmaa.2000.6920. Google Scholar [36] J. Zhang and G. Yan, Three-dimensional Lattice Boltzmann model for the complex Ginzburg-Landau equation, J. Sci. Comput., 60 (2014), 660-683. doi: 10.1007/s10915-013-9811-z. Google Scholar [37] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, Eur. J. Appl. Math., 18 (2007), 607-630. doi: 10.1017/S0956792507007140. Google Scholar [38] R. Zhou, X. Wang and X. Tang, On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation $AX+XB = C$, East Asian J. Appl. Math., 7 (2017), 55-69. doi: 10.4208/eajam.190716.051116a. Google Scholar [39] Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, 1991. Google Scholar

show all references

References:
 [1] F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, Proc. Sympos. Appl. Math., 17 (1965), 24-49. Google Scholar [2] C. Bu, On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation, J. Austral. Math. Soc. Ser. B., 36 (1995), 313-324. doi: 10.1017/S0334270000010468. Google Scholar [3] Z. Cao, B. Guo and B. Wang, Global existence theory for the two-dimensional derivative Ginzburg-Landau equation, Chin. Sci. Bull., 43 (1998), 393-395. doi: 10.1007/BF02883716. Google Scholar [4] Z. Chen, Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity, Numer. Math., 76 (1997), 323-353. doi: 10.1007/s002110050266. Google Scholar [5] Z. Chen and S. Dai, Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity, SIAM J. Numer. Anal., 38 (2001), 1961-1985. doi: 10.1137/S0036142998349102. Google Scholar [6] C. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1998), 279-309. doi: 10.1088/0951-7715/1/2/001. Google Scholar [7] Q. Du, M. D. Gunzburger and J. S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34 (1992), 54-81. doi: 10.1137/1034003. Google Scholar [8] Q. Du, Finite element methods for the time-dependent Ginzberg-Landau model of superconductivity, Comput. Math. Appl., 27 (1994), 119-133. doi: 10.1016/0898-1221(94)90091-4. Google Scholar [9] Q. Du, Discrete gauge invariant approximations of a time dependent Ginzberg-Landau model of superconductivity, Math. Comput., 67 (1998), 965-986. doi: 10.1090/S0025-5718-98-00954-5. Google Scholar [10] L. C. Evans, Partial Differential Equations, Providence, American Mathematical Society, Rhode Island, 1998. doi: 10.1090/gsm/019. Google Scholar [11] H. Gao, B. Li and W. Sun, Optimal error estimates of linearized Crank-Nicolson Galerkin FEMs for the time-dependent Ginzburg-Landau equations in superconductivity, SIAM J. Numer. Anal., 52 (2014), 1183-1202. doi: 10.1137/130918678. Google Scholar [12] H. Gao and J. Duan, Asymptotic for the generalized two dimensional Ginzburg-Landau equation, J. Math. Anal. Appl., 247 (2000), 198-216. doi: 10.1006/jmaa.2000.6848. Google Scholar [13] J. M. Gnidaglia and B. Heron, Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28 (1987), 282-304. doi: 10.1016/0167-2789(87)90020-0. Google Scholar [14] L. Kong, J Hong, L. Ji and P. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Eq., 31 (2015), 1814-1843. doi: 10.1002/num.21969. Google Scholar [15] A. Y. Kolesov and N. K. Rosov, Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain, Theor. Math. Phys., 125 (2000), 1476-1488. doi: 10.1007/BF02551008. Google Scholar [16] L. Kuang, L. Kong, L. Wang and X. Zheng, The splitting high-order compact scheme for two-dimensional Ginzburg-Landau equation, J Jianxi Normal Univer., 40 (2017), 35-38. Google Scholar [17] S. K. Lele, Compact finite difference schemes with spectral-like solution, J. Comput. Phys., 103 (1992), 16-42. doi: 10.1016/0021-9991(92)90324-R. Google Scholar [18] C. D. Levemore and M. Oliver, The complex Ginzburg-Landau equation as a model problem, Lect. Appl. Math., 31 (1996), 141-190. Google Scholar [19] J. Li, High order compact ADI methods for the parabolic equations, Comput. Math. Appl., 52 (2006), 1343-1356. doi: 10.1016/j.camwa.2006.11.010. Google Scholar [20] Y. Li and B. Guo, Global existence of solutions to the derivative Ginzburg-Landau equation, J. Math. Anal. Appl., 249 (2000), 412-432. doi: 10.1006/jmaa.2000.6880. Google Scholar [21] B. Li, Convergence of a decoupled mixed FEM for the dynamic Ginzberg-Landau equations in nonsmooth domains with incompatible initial, Calclo, 54 (2017), 1441-1480. doi: 10.1007/s10092-017-0237-0. Google Scholar [22] B. Li and Z. Zhang, Mathematical and numerical analysis of time-dependent Ginzburg-Landau equations in nonconvex polygons based on hodge decomposition, Math. Comput., 86 (2017), 1579-1608. doi: 10.1090/mcom/3177. Google Scholar [23] S. Lü and Q. Lu, Exponential attractor for the 3D Ginzburg-Landau type equation, Nonlin. Anal., 67 (2007), 3116-3135. doi: 10.1016/j.na.2006.10.005. Google Scholar [24] S. Lü, Q. Lu and E. H. Twizell, Fourier spectral approximation to long-time behavior of the derivative three-dimensional Ginzburg-Landau equation, J. Comput. Appl. Math., 198 (2007), 167-186. doi: 10.1016/j.cam.2005.11.028. Google Scholar [25] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D., 41 (1990), 232-252. doi: 10.1016/0167-2789(90)90125-9. Google Scholar [26] Y. V. S. S. Sanyasiraju and N. Mishra, Spectral resolution exponential compact higher order scheme for convection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 197 (2008), 4737-4744. doi: 10.1016/j.cma.2008.06.013. Google Scholar [27] A Shokri and F Afshari, High-order compact ADI method using predictor-corrector scheme for 2D complex Ginzburg-Landau equation, Comput. Phys. Commun., 197 (2015), 43-50. doi: 10.1016/j.cpc.2015.08.005. Google Scholar [28] B. Wang, Existence of time periodic solutions for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 232 (1999), 394-412. doi: 10.1006/jmaa.1999.6283. Google Scholar [29] H. Wang, An efficient Chebyshev-Tau spectral method for Ginzburg-Landau-Schrödinger equations, Comput. Phys. Comms., 181 (2010), 325-340. doi: 10.1016/j.cpc.2009.10.007. Google Scholar [30] L. Wang, Y. Zhou and L. Fu, The compact and modified ADI scheme for Schrödinger equations, J. Jiangxi Normal Univer., 40 (2016), 515-519. Google Scholar [31] T. Wang and B. Guo, Analysis of some finite difference schemes for two-dimensional Ginzburg-Landau equation, Numer. Meth. Part. Diff. Eq., 27 (2011), 1340-1363. doi: 10.1002/num.20588. Google Scholar [32] P. Wang and C. Huang, An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg-Landau equation, BIT Numer. Math., 58 (2018), 783-805. doi: 10.1007/s10543-018-0698-9. Google Scholar [33] X. Wang, W. Li and L. Mao, On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation $AX+XB = C$., Comput. Math. with Appl., 66 (2013), 2352-2361. doi: 10.1016/j.camwa.2013.09.011. Google Scholar [34] Q. Xu and Q. Chang, Difference methods for computing the Ginzburg-Landau equation in two dimensions, Numer. Meth. Part. Diff. Eq., 27 (2011), 507-528. doi: 10.1002/num.20535. Google Scholar [35] M. Zhan, Existence of periodic solutions for Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 249 (2000), 614-625. doi: 10.1006/jmaa.2000.6920. Google Scholar [36] J. Zhang and G. Yan, Three-dimensional Lattice Boltzmann model for the complex Ginzburg-Landau equation, J. Sci. Comput., 60 (2014), 660-683. doi: 10.1007/s10915-013-9811-z. Google Scholar [37] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, Eur. J. Appl. Math., 18 (2007), 607-630. doi: 10.1017/S0956792507007140. Google Scholar [38] R. Zhou, X. Wang and X. Tang, On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation $AX+XB = C$, East Asian J. Appl. Math., 7 (2017), 55-69. doi: 10.4208/eajam.190716.051116a. Google Scholar [39] Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, 1991. Google Scholar
The evolution of mass against time by Scheme 1
The error of numerical solution in real part against time by Scheme 1 in sense of $L_2$ (Left) and $L_\infty$ (Right)
Plots of the numerical solution at different times. Real part (Left), imaginary part (Right)
The evolution of mass against time of Gaussian pulse by Scheme 1
Plots of $|U_{j,k}^n|$ at different times
Plots of $|U_{j,k}^n|$ with different $\gamma$ at $t = 2$ (left) and $t = 4$ (right)
Plots of $|U_{j,k}^n|$ with different $\beta$ at $t = 2.5$ (left) and $t = 5$ (right)
Spatial convergent test and comparison among all schemes with $\tau = 0.001$
 Scheme $h$ $\|e_{re}^n\|_2$ order $\|e_{re}^n\|_\infty$ order CPU (s) 1 $1$ $2.20\times 10^{-2}$ - $4.93\times 10^{-3}$ - $2.1$ $0.5$ $1.33\times 10^{-3}$ $4.05$ $3.04\times 10^{-4}$ $4.02$ $7.9$ $0.25$ $8.76\times 10^{-5}$ $3.92$ $2.09\times 10^{-5}$ $3.86$ $39$ 2 $1$ $2.20\times 10^{-2}$ - $4.93\times 10^{-3}$ - $1.7$ $0.5$ $1.33\times 10^{-3}$ $4.05$ $3.04\times 10^{-4}$ $4.02$ $6.8$ $0.25$ $7.89\times 10^{-5}$ $4.08$ $1.86\times 10^{-5}$ $4.03$ $34$ 3 $1$ $3.82\times 10^{-1}$ - $8.20\times 10^{-2}$ - $1.1$ $0.5$ $9.61\times 10^{-2}$ $1.99$ $2.23\times 10^{-2}$ $1.88$ $4.4$ $0.25$ $2.41\times 10^{-2}$ $1.98$ $5.66\times 10^{-3}$ $2.00$ $25$ 4 $1$ $3.82\times 10^{-1}$ - $8.20\times 10^{-2}$ - $1.1$ $0.5$ $9.60\times 10^{-2}$ $1.99$ $2.22\times 10^{-2}$ $1.88$ $4.5$ $0.25$ $2.40\times 10^{-2}$ $2.00$ $5.66\times 10^{-3}$ $2.00$ $25$
 Scheme $h$ $\|e_{re}^n\|_2$ order $\|e_{re}^n\|_\infty$ order CPU (s) 1 $1$ $2.20\times 10^{-2}$ - $4.93\times 10^{-3}$ - $2.1$ $0.5$ $1.33\times 10^{-3}$ $4.05$ $3.04\times 10^{-4}$ $4.02$ $7.9$ $0.25$ $8.76\times 10^{-5}$ $3.92$ $2.09\times 10^{-5}$ $3.86$ $39$ 2 $1$ $2.20\times 10^{-2}$ - $4.93\times 10^{-3}$ - $1.7$ $0.5$ $1.33\times 10^{-3}$ $4.05$ $3.04\times 10^{-4}$ $4.02$ $6.8$ $0.25$ $7.89\times 10^{-5}$ $4.08$ $1.86\times 10^{-5}$ $4.03$ $34$ 3 $1$ $3.82\times 10^{-1}$ - $8.20\times 10^{-2}$ - $1.1$ $0.5$ $9.61\times 10^{-2}$ $1.99$ $2.23\times 10^{-2}$ $1.88$ $4.4$ $0.25$ $2.41\times 10^{-2}$ $1.98$ $5.66\times 10^{-3}$ $2.00$ $25$ 4 $1$ $3.82\times 10^{-1}$ - $8.20\times 10^{-2}$ - $1.1$ $0.5$ $9.60\times 10^{-2}$ $1.99$ $2.22\times 10^{-2}$ $1.88$ $4.5$ $0.25$ $2.40\times 10^{-2}$ $2.00$ $5.66\times 10^{-3}$ $2.00$ $25$
Temporal convergent test and comparison among all schemes with $h_x = h_y = 6/80$
 Scheme $\tau$ $\|e_{re}^n\|_2$ order $\|e_{re}^n\|_\infty$ order CPU(s) 1 $0.1$ $4.18\times 10^{-1}$ - $9.85\times 10^{-2}$ - $17.9$ $0.05$ $1.07\times 10^{-1}$ $1.97$ $2.51\times 10^{-2}$ $1.97$ $24.2$ $0.025$ $2.68\times 10^{-2}$ $1.99$ $6.32\times 10^{-3}$ $1.99$ $39.9$ $0.0125$ $6.71\times 10^{-3}$ $2.00$ $1.58\times 10^{-3}$ $2.00$ $72.1$ 2 $0.1$ $2.29\times 10^{-1}$ - $5.39\times 10^{-1}$ - $14.2$ $0.05$ $6.15\times 10^{-2}$ $1.89$ $1.45\times 10^{-2}$ $1.89$ $23.6$ $0.025$ $1.59\times 10^{-2}$ $1.95$ $3.74\times 10^{-3}$ $1.95$ $39.0$ $0.0125$ $4.02\times 10^{-3}$ $1.98$ $9.47\times 10^{-4}$ $1.98$ $61.9$ 3 $0.1$ $4.18\times 10^{-1}$ - $9.85\times 10^{-2}$ - $16.1$ $0.05$ $1.07\times 10^{-2}$ $1.97$ $2.51\times 10^{-2}$ $1.97$ $21.4$ $0.025$ $2.67\times 10^{-2}$ $2.00$ $6.28\times 10^{-3}$ $2.00$ $36.0$ $0.0125$ $6.81\times 10^{-3}$ $1.97$ $1.61\times 10^{-3}$ $1.97$ $58.2$ 4 $0.1$ $2.28\times 10^{-1}$ - $5.38\times 10^{-2}$ - $11.8$ $0.05$ $6.11\times 10^{-2}$ $1.90$ $1.44\times 10^{-2}$ $1.90$ $19.0$ $0.025$ $1.54\times 10^{-2}$ $1.98$ $3.64\times 10^{-3}$ $1.98$ $35.3$ $0.0125$ $3.99\times 10^{-3}$ $1.95$ $9.40\times 10^{-4}$ $1.95$ $50.3$
 Scheme $\tau$ $\|e_{re}^n\|_2$ order $\|e_{re}^n\|_\infty$ order CPU(s) 1 $0.1$ $4.18\times 10^{-1}$ - $9.85\times 10^{-2}$ - $17.9$ $0.05$ $1.07\times 10^{-1}$ $1.97$ $2.51\times 10^{-2}$ $1.97$ $24.2$ $0.025$ $2.68\times 10^{-2}$ $1.99$ $6.32\times 10^{-3}$ $1.99$ $39.9$ $0.0125$ $6.71\times 10^{-3}$ $2.00$ $1.58\times 10^{-3}$ $2.00$ $72.1$ 2 $0.1$ $2.29\times 10^{-1}$ - $5.39\times 10^{-1}$ - $14.2$ $0.05$ $6.15\times 10^{-2}$ $1.89$ $1.45\times 10^{-2}$ $1.89$ $23.6$ $0.025$ $1.59\times 10^{-2}$ $1.95$ $3.74\times 10^{-3}$ $1.95$ $39.0$ $0.0125$ $4.02\times 10^{-3}$ $1.98$ $9.47\times 10^{-4}$ $1.98$ $61.9$ 3 $0.1$ $4.18\times 10^{-1}$ - $9.85\times 10^{-2}$ - $16.1$ $0.05$ $1.07\times 10^{-2}$ $1.97$ $2.51\times 10^{-2}$ $1.97$ $21.4$ $0.025$ $2.67\times 10^{-2}$ $2.00$ $6.28\times 10^{-3}$ $2.00$ $36.0$ $0.0125$ $6.81\times 10^{-3}$ $1.97$ $1.61\times 10^{-3}$ $1.97$ $58.2$ 4 $0.1$ $2.28\times 10^{-1}$ - $5.38\times 10^{-2}$ - $11.8$ $0.05$ $6.11\times 10^{-2}$ $1.90$ $1.44\times 10^{-2}$ $1.90$ $19.0$ $0.025$ $1.54\times 10^{-2}$ $1.98$ $3.64\times 10^{-3}$ $1.98$ $35.3$ $0.0125$ $3.99\times 10^{-3}$ $1.95$ $9.40\times 10^{-4}$ $1.95$ $50.3$
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