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  December 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146
References:
[1]

T. BenjaminJ. 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.

[2]

D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.

[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.

[4]

I. DagB. 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.

[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.

[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.

[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.

[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.

[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.

[10]

C. LuW. 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.

[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.

[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.

[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.

[14]

K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Berlin: Springer-Verlag, 2006.

[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.

[17]

Y. Sun and M. Qin, A multi-symplectic scheme for RLW equation, J. Comput. Math., 22 (2004), 611-621. 

[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.

[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.

[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.

[21]

J. CaiC. 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.

[22]

Y. GongJ. 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.

[23]

J. HongL. Ji and Z. Liu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Appl. Numer. Math., 127 (2018), 164-178. 

[24]

Q. HongY. 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.

[25]

Q. HongY. 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.

[26]

L. KongJ. HongL. 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.

[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.

[28]

T. WangB. 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.

[29]

X. QianH. 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.

[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.

[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.

[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.

[33]

J. CaiY. 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.

[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.

[35]

T. WangL. 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.

[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.

[37]

S. BadiaF. 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.

[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.

[39]

Y. GongJ. 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.

[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.

[41]

J. ZhaoX. YangY. 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.

[42]

Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted.

[43]

Y. GongQ. WangY. 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.

[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.

show all references

References:
[1]

T. BenjaminJ. 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.

[2]

D. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.

[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.

[4]

I. DagB. 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.

[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.

[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.

[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.

[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.

[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.

[10]

C. LuW. 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.

[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.

[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.

[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.

[14]

K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, 2010. doi: 10.1007/978-3-642-01777-3.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Berlin: Springer-Verlag, 2006.

[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.

[17]

Y. Sun and M. Qin, A multi-symplectic scheme for RLW equation, J. Comput. Math., 22 (2004), 611-621. 

[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.

[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.

[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.

[21]

J. CaiC. 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.

[22]

Y. GongJ. 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.

[23]

J. HongL. Ji and Z. Liu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Appl. Numer. Math., 127 (2018), 164-178. 

[24]

Q. HongY. 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.

[25]

Q. HongY. 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.

[26]

L. KongJ. HongL. 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.

[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.

[28]

T. WangB. 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.

[29]

X. QianH. 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.

[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.

[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.

[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.

[33]

J. CaiY. 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.

[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.

[35]

T. WangL. 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.

[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.

[37]

S. BadiaF. 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.

[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.

[39]

Y. GongJ. 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.

[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.

[41]

J. ZhaoX. YangY. 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.

[42]

Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted.

[43]

Y. GongQ. WangY. 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.

[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.

Figure 1.  The accuracy of numerical solutions in $ L^2 $ and $ L^{\infty} $ errors of the four schemes with mesh size $ \tau = h $
Figure 2.  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] $
Figure 3.  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 $
Figure 4.  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 $
Figure 5.  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 $
Figure 6.  Initial and undulation profiles with gentle $ d = 2 $ (top) and $ d = 5 $ (bottom) at different times using the scheme LILF
Figure 7.  (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
Table 1.  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
Table 2.  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
Table 3.  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
Table 4.  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
Table 5.  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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