# American Institute of Mathematical Sciences

December  2019, 24(12): 6445-6464. doi: 10.3934/dcdsb.2019146

## Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

 1 Graduate School of China Academy of Engineering Physics, Beijing 100088, China 2 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 3 College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

* Corresponding author: gongyuezheng@nuaa.edu.cn

Received  November 2018 Revised  January 2019 Published  July 2019

In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

Citation: Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146
##### References:
 [1] T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R Soc. Lond. A, 227 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar [2] D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. doi: 10.1017/S0022112066001678. Google Scholar [3] P. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc., 85 (1979), 143-160. doi: 10.1017/S0305004100055572. Google Scholar [4] I. Dag, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput., 159 (2004), 373-389. doi: 10.1016/j.amc.2003.10.020. Google Scholar [5] M. Dehghan and R. Salehi, The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas, Comput. Phys. Commun., 182 (2011), 2540-2549. doi: 10.1016/j.cpc.2011.07.018. Google Scholar [6] A. Dogan, Numerical solution of RLW equation using linear finite elements within Galerkin's method, Appl. Math. Model., 26 (2002), 771-783. doi: 10.1016/S0307-904X(01)00084-1. Google Scholar [7] Y. Gao and L. Mei, Mixed Galerkin finite element methods for modified regularized long-wave equation, Appl. Math. Comput., 258 (2015), 267-281. doi: 10.1016/j.amc.2015.02.012. Google Scholar [8] H. Gu and N. Chen, Least-squares mixed finite element methods for the RLW equations, Numer. Method Partial Differential Equation, 24 (2008), 749-758. doi: 10.1002/num.20285. Google Scholar [9] B. Guo and W. Cao, The Fourier pseudospectral method with a restrain operator for the RLW equation, J. Comput. Phys., 74 (1988), 110-126. doi: 10.1016/0021-9991(88)90072-1. Google Scholar [10] C. Lu, W. Huang and J. Qiu, An adaptive moving mesh finite element solution of the Regularized Long Wave equation, J. Sci. Comput., 74 (2018), 122-144. doi: 10.1007/s10915-017-0427-6. Google Scholar [11] Z. Luo and R. Liu, Mixed finite element method analysis and numerical solitary for the RLW equation, SIAM J. Numer. Anal., 36 (1999), 89-104. doi: 10.1137/S0036142996312999. Google Scholar [12] L. Mei and Y. Chen, Numerical solutions of RLW equation using Galerkin method with extrapolation techniques, Comput. Phys. Commun., 183 (2012), 1609-1616. doi: 10.1016/j.cpc.2012.02.029. Google Scholar [13] S. Zaki, Solitary waves of the splitted RLW equation, Comput. Phys. Comm., 138 (2001), 80-91. doi: 10.1016/S0010-4655(01)00200-4. Google Scholar [14] K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3. Google Scholar [15] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Berlin: Springer-Verlag, 2006. Google Scholar [16] C. Bubb and M. Piggot, Geometric integration and its application, Handbook of Numerical Analysis, Vol. XI, 35–139, Handb. Numer. Anal., XI, North-Holland, Amsterdam, 2003. Google Scholar [17] Y. Sun and M. Qin, A multi-symplectic scheme for RLW equation, J. Comput. Math., 22 (2004), 611-621. Google Scholar [18] J. Cai, Multi-symplectic numerical method for the regularized long-wave equation, Comput. Phys. Commun., 180 (2009), 1821-1831. doi: 10.1016/j.cpc.2009.05.009. Google Scholar [19] J. Cai, A new explicit multi-symplectic scheme for the regularized long-wave equation, J. Math. Phys., 50 (2009), 013535, 16pp. doi: 10.1063/1.3068404. Google Scholar [20] Q. Hong, Y. Wang and Y. Gong, Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation, arXiv: 1806.08948.Google Scholar [21] J. Cai, C. Bai and H. Zhang, Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations, J. Comput. Phys., 374 (2018), 281-299. doi: 10.1016/j.jcp.2018.07.050. Google Scholar [22] Y. Gong, J. Cai and Y. Wang, Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), 80-102. doi: 10.1016/j.jcp.2014.09.001. Google Scholar [23] J. Hong, L. Ji and Z. Liu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Appl. Numer. Math., 127 (2018), 164-178. Google Scholar [24] Q. Hong, Y. Wang and Q. Du, Two new energy-preserving algorithms for generalized fifth-order KdV equation, Adv. Appl. Math. Mech., 9 (2017), 1206-1224. doi: 10.4208/aamm.OA-2016-0044. Google Scholar [25] Q. Hong, Y. Wang and J. Wang, Optimal error estimate of a linear Fourier pseudo-spectral scheme for the two dimensional Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 468 (2018), 817-838. doi: 10.1016/j.jmaa.2018.08.045. Google Scholar [26] L. Kong, J. Hong, L. Ji and P. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations, 31 (2015), 1814-1843. doi: 10.1002/num.21969. Google Scholar [27] Z. Sun and D. Zhao, On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl., 59 (2010), 3286-3300. doi: 10.1016/j.camwa.2010.03.012. Google Scholar [28] T. Wang, B. Guo and Q. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243 (2013), 382-399. doi: 10.1016/j.jcp.2013.03.007. Google Scholar [29] X. Qian, H. Fu and S. Song, Structure-preserving wavelet algorithms for the nonlinear Dirac model, Adv. Appl. Math. Mech., 9 (2017), 964-989. doi: 10.4208/aamm.2016.m1463. Google Scholar [30] J. Wang and Y. Wang, Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations, Int. J. Comput. Math., 95 (2018), 1583-1608. doi: 10.1080/00207160.2017.1322692. Google Scholar [31] D. Furihata, Finite difference schemes for $\frac{\partial u}{\partial t} = (\frac{\partial}{\partial x})^{\alpha}\frac{\delta G}{\delta u}$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377. Google Scholar [32] D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447. doi: 10.1006/jcph.2001.6775. Google Scholar [33] J. Cai, Y. Gong and H. Liang, Novel implicit/explicit local conservative scheme for the regularized long-wave equation and convergence analysis, J. Math. Anal. Appl., 447 (2017), 17-31. doi: 10.1016/j.jmaa.2016.09.047. Google Scholar [34] J. Cai and Q. Hong, Efficient local structure-preserving schemes for the RLW-type equation, Numer. Methods Partial Differential Equations, 33 (2017), 1678-1691. doi: 10.1002/num.22162. Google Scholar [35] T. Wang, L. Zhang and F. Chen, Conservative schemes for the symmetric regularized long wave equations, Appl. Math. Comput., 190 (2007), 1063-1080. doi: 10.1016/j.amc.2007.01.105. Google Scholar [36] M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for pdes, SIAM J. Sci. Comput., 33 (2011), 2318-2340. doi: 10.1137/100810174. Google Scholar [37] S. Badia, F. Guillen-Gonzalez and J. Gutierrez-Santacreu, Finite element approximation of nematic liquid crystal flows using a saddle-point structure, J. Comput. Phys., 230 (2011), 1686-1706. doi: 10.1016/j.jcp.2010.11.033. Google Scholar [38] F. Guillen and G. Tierra, Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68 (2014), 821-846. doi: 10.1016/j.camwa.2014.07.014. Google Scholar [39] Y. Gong, J. Zhao and Q. Wang, Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation, Adv. Comput. Math., 44 (2018), 1573-1600. doi: 10.1007/s10444-018-9597-5. Google Scholar [40] X. Yang and D. Han, Linearly first-and second-order, unconditionally energy stable schemes for the phase field crystal equation, J. Comput. Phys., 333 (2017), 1116-1134. doi: 10.1016/j.jcp.2016.10.020. Google Scholar [41] J. Zhao, X. Yang, Y. Gong and Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg., 318 (2017), 803-825. doi: 10.1016/j.cma.2017.01.031. Google Scholar [42] Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted.Google Scholar [43] Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudospectral method for the nonlinear Schrodinger equation, J. Comput. Phys., 328 (2017), 354-370. doi: 10.1016/j.jcp.2016.10.022. Google Scholar [44] J. Cai, Some linearly and nonlinearly implicit schemes for the numerical solutions of the regularized long-wave equation, Appl. Math. Comput., 217 (2011), 9948-9955. doi: 10.1016/j.amc.2011.04.040. Google Scholar

show all references

##### References:
 [1] T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R Soc. Lond. A, 227 (1972), 47-78. doi: 10.1098/rsta.1972.0032. Google Scholar [2] D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330. doi: 10.1017/S0022112066001678. Google Scholar [3] P. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Camb. Phil. Soc., 85 (1979), 143-160. doi: 10.1017/S0305004100055572. Google Scholar [4] I. Dag, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput., 159 (2004), 373-389. doi: 10.1016/j.amc.2003.10.020. Google Scholar [5] M. Dehghan and R. Salehi, The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas, Comput. Phys. Commun., 182 (2011), 2540-2549. doi: 10.1016/j.cpc.2011.07.018. Google Scholar [6] A. Dogan, Numerical solution of RLW equation using linear finite elements within Galerkin's method, Appl. Math. Model., 26 (2002), 771-783. doi: 10.1016/S0307-904X(01)00084-1. Google Scholar [7] Y. Gao and L. Mei, Mixed Galerkin finite element methods for modified regularized long-wave equation, Appl. Math. Comput., 258 (2015), 267-281. doi: 10.1016/j.amc.2015.02.012. Google Scholar [8] H. Gu and N. Chen, Least-squares mixed finite element methods for the RLW equations, Numer. Method Partial Differential Equation, 24 (2008), 749-758. doi: 10.1002/num.20285. Google Scholar [9] B. Guo and W. Cao, The Fourier pseudospectral method with a restrain operator for the RLW equation, J. Comput. Phys., 74 (1988), 110-126. doi: 10.1016/0021-9991(88)90072-1. Google Scholar [10] C. Lu, W. Huang and J. Qiu, An adaptive moving mesh finite element solution of the Regularized Long Wave equation, J. Sci. Comput., 74 (2018), 122-144. doi: 10.1007/s10915-017-0427-6. Google Scholar [11] Z. Luo and R. Liu, Mixed finite element method analysis and numerical solitary for the RLW equation, SIAM J. Numer. Anal., 36 (1999), 89-104. doi: 10.1137/S0036142996312999. Google Scholar [12] L. Mei and Y. Chen, Numerical solutions of RLW equation using Galerkin method with extrapolation techniques, Comput. Phys. Commun., 183 (2012), 1609-1616. doi: 10.1016/j.cpc.2012.02.029. Google Scholar [13] S. Zaki, Solitary waves of the splitted RLW equation, Comput. Phys. Comm., 138 (2001), 80-91. doi: 10.1016/S0010-4655(01)00200-4. Google Scholar [14] K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3. Google Scholar [15] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Berlin: Springer-Verlag, 2006. Google Scholar [16] C. Bubb and M. Piggot, Geometric integration and its application, Handbook of Numerical Analysis, Vol. XI, 35–139, Handb. Numer. Anal., XI, North-Holland, Amsterdam, 2003. Google Scholar [17] Y. Sun and M. Qin, A multi-symplectic scheme for RLW equation, J. Comput. Math., 22 (2004), 611-621. Google Scholar [18] J. Cai, Multi-symplectic numerical method for the regularized long-wave equation, Comput. Phys. Commun., 180 (2009), 1821-1831. doi: 10.1016/j.cpc.2009.05.009. Google Scholar [19] J. Cai, A new explicit multi-symplectic scheme for the regularized long-wave equation, J. Math. Phys., 50 (2009), 013535, 16pp. doi: 10.1063/1.3068404. Google Scholar [20] Q. Hong, Y. Wang and Y. Gong, Optimal error estimate of two linear and momentum-preserving Fourier pseudo-spectral schemes for the RLW equation, arXiv: 1806.08948.Google Scholar [21] J. Cai, C. Bai and H. Zhang, Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations, J. Comput. Phys., 374 (2018), 281-299. doi: 10.1016/j.jcp.2018.07.050. Google Scholar [22] Y. Gong, J. Cai and Y. Wang, Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), 80-102. doi: 10.1016/j.jcp.2014.09.001. Google Scholar [23] J. Hong, L. Ji and Z. Liu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Appl. Numer. Math., 127 (2018), 164-178. Google Scholar [24] Q. Hong, Y. Wang and Q. Du, Two new energy-preserving algorithms for generalized fifth-order KdV equation, Adv. Appl. Math. Mech., 9 (2017), 1206-1224. doi: 10.4208/aamm.OA-2016-0044. Google Scholar [25] Q. Hong, Y. Wang and J. Wang, Optimal error estimate of a linear Fourier pseudo-spectral scheme for the two dimensional Klein-Gordon-Schrödinger equations, J. Math. Anal. Appl., 468 (2018), 817-838. doi: 10.1016/j.jmaa.2018.08.045. Google Scholar [26] L. Kong, J. Hong, L. Ji and P. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations, 31 (2015), 1814-1843. doi: 10.1002/num.21969. Google Scholar [27] Z. Sun and D. Zhao, On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl., 59 (2010), 3286-3300. doi: 10.1016/j.camwa.2010.03.012. Google Scholar [28] T. Wang, B. Guo and Q. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243 (2013), 382-399. doi: 10.1016/j.jcp.2013.03.007. Google Scholar [29] X. Qian, H. Fu and S. Song, Structure-preserving wavelet algorithms for the nonlinear Dirac model, Adv. Appl. Math. Mech., 9 (2017), 964-989. doi: 10.4208/aamm.2016.m1463. Google Scholar [30] J. Wang and Y. Wang, Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations, Int. J. Comput. Math., 95 (2018), 1583-1608. doi: 10.1080/00207160.2017.1322692. Google Scholar [31] D. Furihata, Finite difference schemes for $\frac{\partial u}{\partial t} = (\frac{\partial}{\partial x})^{\alpha}\frac{\delta G}{\delta u}$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377. Google Scholar [32] D. Furihata, Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, J. Comput. Phys., 171 (2001), 425-447. doi: 10.1006/jcph.2001.6775. Google Scholar [33] J. Cai, Y. Gong and H. Liang, Novel implicit/explicit local conservative scheme for the regularized long-wave equation and convergence analysis, J. Math. Anal. Appl., 447 (2017), 17-31. doi: 10.1016/j.jmaa.2016.09.047. Google Scholar [34] J. Cai and Q. Hong, Efficient local structure-preserving schemes for the RLW-type equation, Numer. Methods Partial Differential Equations, 33 (2017), 1678-1691. doi: 10.1002/num.22162. Google Scholar [35] T. Wang, L. Zhang and F. Chen, Conservative schemes for the symmetric regularized long wave equations, Appl. Math. Comput., 190 (2007), 1063-1080. doi: 10.1016/j.amc.2007.01.105. Google Scholar [36] M. Dahlby and B. Owren, A general framework for deriving integral preserving numerical methods for pdes, SIAM J. Sci. Comput., 33 (2011), 2318-2340. doi: 10.1137/100810174. Google Scholar [37] S. Badia, F. Guillen-Gonzalez and J. Gutierrez-Santacreu, Finite element approximation of nematic liquid crystal flows using a saddle-point structure, J. Comput. Phys., 230 (2011), 1686-1706. doi: 10.1016/j.jcp.2010.11.033. Google Scholar [38] F. Guillen and G. Tierra, Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68 (2014), 821-846. doi: 10.1016/j.camwa.2014.07.014. Google Scholar [39] Y. Gong, J. Zhao and Q. Wang, Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation, Adv. Comput. Math., 44 (2018), 1573-1600. doi: 10.1007/s10444-018-9597-5. Google Scholar [40] X. Yang and D. Han, Linearly first-and second-order, unconditionally energy stable schemes for the phase field crystal equation, J. Comput. Phys., 333 (2017), 1116-1134. doi: 10.1016/j.jcp.2016.10.020. Google Scholar [41] J. Zhao, X. Yang, Y. Gong and Q. Wang, A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals, Comput. Methods Appl. Mech. Engrg., 318 (2017), 803-825. doi: 10.1016/j.cma.2017.01.031. Google Scholar [42] Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted.Google Scholar [43] Y. Gong, Q. Wang, Y. Wang and J. Cai, A conservative Fourier pseudospectral method for the nonlinear Schrodinger equation, J. Comput. Phys., 328 (2017), 354-370. doi: 10.1016/j.jcp.2016.10.022. Google Scholar [44] J. Cai, Some linearly and nonlinearly implicit schemes for the numerical solutions of the regularized long-wave equation, Appl. Math. Comput., 217 (2011), 9948-9955. doi: 10.1016/j.amc.2011.04.040. Google Scholar
The accuracy of numerical solutions in $L^2$ and $L^{\infty}$ errors of the four schemes with mesh size $\tau = h$
Comparison of $L^2$ and $L^{\infty}$ errors in numerical solutions and CPU time(s) at $T = 1$, where $c = 1/3$ and $x\in[-40,60]$
The errors in mass (left) and energy (right) of the four schemes with $c = 1/3$, $\tau = 0.05$, $h = 0.1$ and $x\in[-60,200]$ until $T = 75$
The evolution of the RLW equation using the scheme LILF with $\sigma = 0.04$ (left), $\sigma = 0.01$ (middle) and $\sigma = 0.001$ (right) at $T = 55$
The errors in mass (left) and energy (right) of the four schemes with $\sigma = 0.01$ $\tau = 0.05$ and $h = 0.05$ and $x\in[-40,100]$ until $T = 55$
Initial and undulation profiles with gentle $d = 2$ (top) and $d = 5$ (bottom) at different times using the scheme LILF
(a) Development of the first undulation from $t = 0$ to $t = 250$ and (b) the behavior of the invariants for $d = 2$ and (c) $d = 5$ by LILF
The invariants and errors of numerical solutions for the scheme FIEP with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 8.291e-5 3.357e-5 8 3.97993 0.42983 1.633e-4 6.721e-5 12 3.97993 0.42983 2.404e-4 9.791e-5 16 3.97993 0.42983 3.138e-4 1.255e-4
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 8.291e-5 3.357e-5 8 3.97993 0.42983 1.633e-4 6.721e-5 12 3.97993 0.42983 2.404e-4 9.791e-5 16 3.97993 0.42983 3.138e-4 1.255e-4
The invariants and errors of numerical solutions for the scheme LIEP with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42979 / / 4 3.97993 0.42979 4.020e-5 1.455e-5 8 3.97993 0.42979 8.265e-5 3.124e-5 12 3.97993 0.42979 1.224e-4 4.673e-5 16 3.97993 0.42979 1.614e-4 6.131e-5
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42979 / / 4 3.97993 0.42979 4.020e-5 1.455e-5 8 3.97993 0.42979 8.265e-5 3.124e-5 12 3.97993 0.42979 1.224e-4 4.673e-5 16 3.97993 0.42979 1.614e-4 6.131e-5
The invariants and errors of numerical solutions for the scheme LICN with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 6.485e-5 2.651e-5 8 3.97993 0.42983 1.270e-4 5.174e-5 12 3.97993 0.42983 1.854e-4 7.413e-5 16 3.97993 0.42983 2.416e-4 9.502e-5
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 6.485e-5 2.651e-5 8 3.97993 0.42983 1.270e-4 5.174e-5 12 3.97993 0.42983 1.854e-4 7.413e-5 16 3.97993 0.42983 2.416e-4 9.502e-5
The invariants and errors of numerical solutions for the scheme LILF with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 1.998e-4 7.882e-5 8 3.97993 0.42983 3.936e-4 1.574e-4 12 3.97993 0.42983 5.842e-4 2.332e-4 16 3.97993 0.42983 7.671e-4 3.021e-4
 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 1.998e-4 7.882e-5 8 3.97993 0.42983 3.936e-4 1.574e-4 12 3.97993 0.42983 5.842e-4 2.332e-4 16 3.97993 0.42983 7.671e-4 3.021e-4
Numerical comparison at $T = 10$ with $c = 0.1$, $\tau = 0.1$ and $-40\leq x\leq 60$.
 Method $h=0.125$ $h=0.0625$ $L^2$ error $L^{\infty}$ error CPU(s) $L^2$ error $L^{\infty}$ error CPU(s) FIEP 2.023e-4 8.298e-5 1.398 1.363e-4 5.520e-5 1.796 LIEP 1.035e-4 3.946e-5 0.993 1.687e-4 6.716e-5 1.268 LICN 1.566e-4 6.316e-5 1.073 9.172e-5 3.545e-5 1.076 LILF 4.896e-4 1.962e-4 1.006 4.241e-4 1.684e-4 1.128 NC-II [34] 2.088e-4 7.532e-5 1.755 1.234e-4 4.236e-5 2.137 AMC-CN [44] 3.944e-4 1.581e-4 1.506 1.828e-4 7.307e-5 1.988 Linear-CN [6] 2.648e-4 1.088e-4 1.046 7.945e-4 2.957e-4 1.108
 Method $h=0.125$ $h=0.0625$ $L^2$ error $L^{\infty}$ error CPU(s) $L^2$ error $L^{\infty}$ error CPU(s) FIEP 2.023e-4 8.298e-5 1.398 1.363e-4 5.520e-5 1.796 LIEP 1.035e-4 3.946e-5 0.993 1.687e-4 6.716e-5 1.268 LICN 1.566e-4 6.316e-5 1.073 9.172e-5 3.545e-5 1.076 LILF 4.896e-4 1.962e-4 1.006 4.241e-4 1.684e-4 1.128 NC-II [34] 2.088e-4 7.532e-5 1.755 1.234e-4 4.236e-5 2.137 AMC-CN [44] 3.944e-4 1.581e-4 1.506 1.828e-4 7.307e-5 1.988 Linear-CN [6] 2.648e-4 1.088e-4 1.046 7.945e-4 2.957e-4 1.108
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