# 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  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. 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. [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. 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. [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. 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. [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. 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. [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. [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. [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. [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. [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. [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. 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. [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. [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. 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. [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. 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. [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. 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. [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. 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. [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. 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. [42] Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted. [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. [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. 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. [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. 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. [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. 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. [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. 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. [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. [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. [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. [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. [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. [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. 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. [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. [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. 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. [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. 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. [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. 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. [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. 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. [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. 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. [42] Y. Gong, Y. Wang and Q. Wang, Linear-implicit conservative schemes based on energy quadratization for Hamiltonian PDEs, submitted. [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. [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.
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
 [1] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [2] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3685-3701. doi: 10.3934/dcdss.2020466 [3] Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006 [4] H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066 [5] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [6] Gianluca Frasca-Caccia, Peter E. Hydon. Locally conservative finite difference schemes for the modified KdV equation. Journal of Computational Dynamics, 2019, 6 (2) : 307-323. doi: 10.3934/jcd.2019015 [7] Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105 [8] Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure and Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141 [9] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [10] Carlos E. Kenig. The method of energy channels for nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6979-6993. doi: 10.3934/dcds.2019240 [11] Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial and Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045 [12] Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 [13] Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021 [14] Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic and Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713 [15] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [16] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 [17] Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024 [18] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [19] Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255 [20] Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems and Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044

2020 Impact Factor: 1.327