Locally conservative finite difference schemes for the modified KdV equation

Gianluca Frasca-Caccia; Peter E. Hydon

doi:10.3934/jcd.2019015

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December 2019, 6(2): 307-323. doi: 10.3934/jcd.2019015

Locally conservative finite difference schemes for the modified KdV equation

Gianluca Frasca-Caccia and Peter E. Hydon ^,

School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK

^* Corresponding author: Peter E. Hydon

Received March 2019 Revised July 2019 Published November 2019

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity.

In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.

Keywords: Finite difference methods, discrete conservation laws, modified KdV equation, energy conservation, momentum conservation.

Mathematics Subject Classification: Primary: 65M06, 37K05; Secondary: 39A14.

Citation: 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

References:

[1]	S. C. Anco, M. Mohiuddin and T. Wolf, Traveling waves and conservation laws for complex mKdV-type equations, Appl. Math. Comput., 219 (2012), 679-698. doi: 10.1016/j.amc.2012.06.061.
[2]	U. M. Ascher and R. I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), 255-269. doi: 10.1016/j.apnum.2003.09.002.
[3]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25 (2005), 83-104. doi: 10.1007/s10915-004-4634-6.
[4]	A. Aydin and B. Karasözen, Multisymplectic box schemes for the complex modified Korteweg-de Vries equation, J. Math. Phys., 51 (2010), 24pp. doi: 10.1063/1.3456068.
[5]	L. Barletti, L. Brugnano, G. Frasca-Caccia and F. Iavernaro, Energy-conserving methods for the nonlinear Schrödinger equation, Appl. Math. Comput., 318 (2018), 3-18. doi: 10.1016/j.amc.2017.04.018.
[6]	T. J. Bridges, Multisymplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190. doi: 10.1017/S0305004196001429.
[7]	T. J. Bridges, P. E. Hydon and J. K. Lawson, Multisymplectic structures and the variational bicomplex, Math. Proc. Cambridge Philos. Soc., 148 (2010), 159-178. doi: 10.1017/S0305004109990259.
[8]	T. J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8.
[9]	T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, 39 (2006), 5287-5320. doi: 10.1088/0305-4470/39/19/S02.
[10]	L. Brugnano and F. Iavernaro, Line Integral Methods for Conservative Problems, Monograph and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b19319.
[11]	B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math., 103 (2006), 197-223. doi: 10.1007/s00211-006-0680-3.
[12]	E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neal, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789. doi: 10.1016/j.jcp.2012.06.022.
[13]	E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Energy-preserving integrators and the structure of B-series, Found. Comput. Math., 10 (2010), 673-693. doi: 10.1007/s10208-010-9073-1.
[14]	D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-2181-2.
[15]	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.
[16]	A. Durán and M. A. López-Marcos, Conservative numerical methods for solitary wave interactions, J. Phys. A, 36 (2003), 7761-7770. doi: 10.1088/0305-4470/36/28/306.
[17]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. Geometric theory, Nonlinearity, 11 (1998), 1547-1567. doi: 10.1088/0951-7715/11/6/008.
[18]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation, IMA J. Numer. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	G. Frasca-Caccia, Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation, AIP Conf. Proc., 2116 (2019). doi: 10.1063/1.5114131.
[20]	G. Frasca-Caccia and P. E. Hydon, Simple bespoke preservation of two conservation laws, IMA J. Numer. Anal., in press. doi: 10.1093/imanum/dry087.
[21]	J. de Frutos and J. M. Sanz-Serna, Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation, Numer. Math., 75 (1997), 421-445. doi: 10.1007/s002110050247.
[22]	D. Furihata, Finite difference schemes for $\partial u/\partial t = (\partial/\partial x)^\alpha\delta G/\delta u$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377.
[23]	D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.
[24]	O. Gonzales, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.
[25]	T. J. Grant, Bespoke finite difference schemes that preserve multiple conservation laws, LMS J. Comput. Math., 18 (2015), 372-403. doi: 10.1112/S1461157015000078.
[26]	T. J. Grant and P. E. Hydon, Characteristics of conservation laws for difference equations, Found. Comput. Math., 13 (2013), 667-692. doi: 10.1007/s10208-013-9151-2.
[27]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010. doi: 10.1007/3-540-30666-8.
[28]	P. E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A, 34 (2001), 10347-10355. doi: 10.1088/0305-4470/34/48/301.
[29]	P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge Monographs on Applied and Computational Mathematics, 27, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139016988.
[30]	P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004), 187-217. doi: 10.1007/s10208-002-0071-9.
[31]	B. A. Kuperschmidt, Discrete Lax equations and differential-difference calculus, Astérisque, (1985), 212pp.
[32]	B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511614118.
[33]	J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys., 199 (1998), 351–395. doi: 10.1007/s002200050505.
[34]	F. McDonald, R. I. McLachlan, B. E. Moore and G. R. W. Quispel, Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations, J. Difference Equ. Appl., 22 (2016), 913-940. doi: 10.1080/10236198.2016.1162161.
[35]	R. I. McLachlan and G. R. W. Quispel, Discrete gradient methods have an energy conservation law, Discrete Contin. Dyn. Syst., 34 (2014), 1099-1104. doi: 10.3934/dcds.2014.34.1099.
[36]	R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363.
[37]	R. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constant of motion, J. Mathematical Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.
[38]	M. Oliver, M. West and C. Wulff, Approximate momentum conservation for spatial semidiscretization of semilinear wave equations, Numer. Math., 97 (2004), 493-535. doi: 10.1007/s00211-003-0488-3.
[39]	P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Mathematical Phys., 18 (1977), 1212-1215. doi: 10.1063/1.523393.
[40]	G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 7pp. doi: 10.1088/1751-8113/41/4/045206.
[41]	G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral, J. Phys. A., 29 (1996), L341–L349. doi: 10.1088/0305-4470/29/13/006.
[42]	D. J. Zhang, S. L. Zhao, Y. Y. Sun and J. Zhou, Solutions to the modified Korteweg-de Vries equation, Rev. Math. Phys., 26 (2014), 42pp. doi: 10.1142/S0129055X14300064.

show all references

References:

[1]	S. C. Anco, M. Mohiuddin and T. Wolf, Traveling waves and conservation laws for complex mKdV-type equations, Appl. Math. Comput., 219 (2012), 679-698. doi: 10.1016/j.amc.2012.06.061.
[2]	U. M. Ascher and R. I. McLachlan, Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), 255-269. doi: 10.1016/j.apnum.2003.09.002.
[3]	U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25 (2005), 83-104. doi: 10.1007/s10915-004-4634-6.
[4]	A. Aydin and B. Karasözen, Multisymplectic box schemes for the complex modified Korteweg-de Vries equation, J. Math. Phys., 51 (2010), 24pp. doi: 10.1063/1.3456068.
[5]	L. Barletti, L. Brugnano, G. Frasca-Caccia and F. Iavernaro, Energy-conserving methods for the nonlinear Schrödinger equation, Appl. Math. Comput., 318 (2018), 3-18. doi: 10.1016/j.amc.2017.04.018.
[6]	T. J. Bridges, Multisymplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190. doi: 10.1017/S0305004196001429.
[7]	T. J. Bridges, P. E. Hydon and J. K. Lawson, Multisymplectic structures and the variational bicomplex, Math. Proc. Cambridge Philos. Soc., 148 (2010), 159-178. doi: 10.1017/S0305004109990259.
[8]	T. J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8.
[9]	T. J. Bridges and S. Reich, Numerical methods for Hamiltonian PDEs, J. Phys. A, 39 (2006), 5287-5320. doi: 10.1088/0305-4470/39/19/S02.
[10]	L. Brugnano and F. Iavernaro, Line Integral Methods for Conservative Problems, Monograph and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b19319.
[11]	B. Cano, Conserved quantities of some Hamiltonian wave equations after full discretization, Numer. Math., 103 (2006), 197-223. doi: 10.1007/s00211-006-0680-3.
[12]	E. Celledoni, V. Grimm, R. I. McLachlan, D. I. McLaren, D. O'Neal, B. Owren and G. R. W. Quispel, Preserving energy resp. dissipation in numerical PDEs using the "average vector field" method, J. Comput. Phys., 231 (2012), 6770-6789. doi: 10.1016/j.jcp.2012.06.022.
[13]	E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Energy-preserving integrators and the structure of B-series, Found. Comput. Math., 10 (2010), 673-693. doi: 10.1007/s10208-010-9073-1.
[14]	D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-2181-2.
[15]	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.
[16]	A. Durán and M. A. López-Marcos, Conservative numerical methods for solitary wave interactions, J. Phys. A, 36 (2003), 7761-7770. doi: 10.1088/0305-4470/36/28/306.
[17]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. Geometric theory, Nonlinearity, 11 (1998), 1547-1567. doi: 10.1088/0951-7715/11/6/008.
[18]	A. Durán and J. M. Sanz-Serna, The numerical integration of relative equilibrium solutions. The nonlinear Schrödinger equation, IMA J. Numer. Anal., 20 (2000), 235-261. doi: 10.1093/imanum/20.2.235.
[19]	G. Frasca-Caccia, Bespoke finite difference methods that preserve two local conservation laws of the modified KdV equation, AIP Conf. Proc., 2116 (2019). doi: 10.1063/1.5114131.
[20]	G. Frasca-Caccia and P. E. Hydon, Simple bespoke preservation of two conservation laws, IMA J. Numer. Anal., in press. doi: 10.1093/imanum/dry087.
[21]	J. de Frutos and J. M. Sanz-Serna, Accuracy and conservation properties in numerical integration: The case of the Korteweg-de Vries equation, Numer. Math., 75 (1997), 421-445. doi: 10.1007/s002110050247.
[22]	D. Furihata, Finite difference schemes for $\partial u/\partial t = (\partial/\partial x)^\alpha\delta G/\delta u$ that inherit energy conservation or dissipation property, J. Comput. Phys., 156 (1999), 181-205. doi: 10.1006/jcph.1999.6377.
[23]	D. Furihata and T. Matsuo, Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011.
[24]	O. Gonzales, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467. doi: 10.1007/BF02440162.
[25]	T. J. Grant, Bespoke finite difference schemes that preserve multiple conservation laws, LMS J. Comput. Math., 18 (2015), 372-403. doi: 10.1112/S1461157015000078.
[26]	T. J. Grant and P. E. Hydon, Characteristics of conservation laws for difference equations, Found. Comput. Math., 13 (2013), 667-692. doi: 10.1007/s10208-013-9151-2.
[27]	E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer, Heidelberg, 2010. doi: 10.1007/3-540-30666-8.
[28]	P. E. Hydon, Conservation laws of partial difference equations with two independent variables, J. Phys. A, 34 (2001), 10347-10355. doi: 10.1088/0305-4470/34/48/301.
[29]	P. E. Hydon, Difference Equations by Differential Equation Methods, Cambridge Monographs on Applied and Computational Mathematics, 27, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139016988.
[30]	P. E. Hydon and E. L. Mansfield, A variational complex for difference equations, Found. Comput. Math., 4 (2004), 187-217. doi: 10.1007/s10208-002-0071-9.
[31]	B. A. Kuperschmidt, Discrete Lax equations and differential-difference calculus, Astérisque, (1985), 212pp.
[32]	B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 14, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511614118.
[33]	J. E. Marsden, G. W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys., 199 (1998), 351–395. doi: 10.1007/s002200050505.
[34]	F. McDonald, R. I. McLachlan, B. E. Moore and G. R. W. Quispel, Travelling wave solutions of multisymplectic discretizations of semi-linear wave equations, J. Difference Equ. Appl., 22 (2016), 913-940. doi: 10.1080/10236198.2016.1162161.
[35]	R. I. McLachlan and G. R. W. Quispel, Discrete gradient methods have an energy conservation law, Discrete Contin. Dyn. Syst., 34 (2014), 1099-1104. doi: 10.3934/dcds.2014.34.1099.
[36]	R. I. McLachlan, G. R. W. Quispel and N. Robidoux, Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), 1021-1045. doi: 10.1098/rsta.1999.0363.
[37]	R. Miura, C. S. Gardner and M. D. Kruskal, Korteweg-de Vries equation and generalizations. Ⅱ. Existence of conservation laws and constant of motion, J. Mathematical Phys., 9 (1968), 1204-1209. doi: 10.1063/1.1664701.
[38]	M. Oliver, M. West and C. Wulff, Approximate momentum conservation for spatial semidiscretization of semilinear wave equations, Numer. Math., 97 (2004), 493-535. doi: 10.1007/s00211-003-0488-3.
[39]	P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Mathematical Phys., 18 (1977), 1212-1215. doi: 10.1063/1.523393.
[40]	G. R. W. Quispel and D. I. McLaren, A new class of energy-preserving numerical integration methods, J. Phys. A, 41 (2008), 7pp. doi: 10.1088/1751-8113/41/4/045206.
[41]	G. R. W. Quispel and G. S. Turner, Discrete gradient methods for solving ODEs numerically while preserving a first integral, J. Phys. A., 29 (1996), L341–L349. doi: 10.1088/0305-4470/29/13/006.
[42]	D. J. Zhang, S. L. Zhao, Y. Y. Sun and J. Zhou, Solutions to the modified Korteweg-de Vries equation, Rev. Math. Phys., 26 (2014), 42pp. doi: 10.1142/S0129055X14300064.

Figure 1. Example of a rectangular stencil for mKdV. PDEs and conservation laws are preserved to second order at the central point $ (x, t) $; densities and fluxes are second-order at $ (x, t-\Delta t/2) $ and $ (x-\Delta x/2, t) $, respectively

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Figure 2. Two-soliton problem for the mKdV equation. Top: Initial condition (dashed line) and solution of $ \mbox{EC}_{10}(0.04) $ with $ \Delta x = 0.1 $, $ \Delta t = 0.025 $ at time $ T = 10 $ (solid line). Bottom: Top of the faster soliton; exact profile (solid line) and solutions of $ \mbox{EC}_{10}(0.04) $ (circles), narrow box (squares), multisymplectic (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses)

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Figure 3. Two-soliton problem for the mKdV equation with $ \Delta x = 0.2 $, $ \Delta t = 0.05 $ at time $ T = 10 $. Top of the soliton: exact profile (solid line) and solutions of $ \mbox{EC}_{10}(0.05) $ (circles), narrow box scheme (squares), multisymplectic scheme (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses)

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Figure 4. Breather problem. Top: Numerical solution given by method $ \mbox{EC}_{8}(2.22) $ with stepsizes $ \Delta x = 0.02 $, $ \Delta t = 0.002 $ at time $ T = 0.4 $. Bottom: exact profile (solid line) and solutions of methods $ \mbox{EC}_{8}(2.22) $ (circles), multisymplectic (diamonds) and $ \mbox{EC}_{10}(0) $ (crosses) at the final time (markers at every tenth point)

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Table 1. Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $ \Delta x\! = \!0.1 $, $ \Delta t\! = \!0.025 $. An asterisk denotes the error that is minimized

Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Sol. Err.	$ \text{Err}_{\phi_1} $	$ \text{Err}_{\phi_2} $	$ \text{Err}_{\phi} $
$ \mbox{EC}_{8}(0) $	1.74e-13	0.0036	5.13e-13	0.3701	-0.51	-0.06	-0.45
$ \mbox{EC}_{8}(1) $	1.33e-13	0.0732	8.01e-13$ ^* $	0.0085$ ^* $	0	-0.01	0.01
$ \mbox{EC}_{8}(-0.05) $	6.22e-14$ ^* $	1.81e-04$ ^* $	4.65e-13	0.3857	-0.53	-0.07	-0.46
$ \mbox{MC}_{8}(0) $	2.13e-13	3.69e-13	0.0632	0.2396	-0.32	-0.04	-0.28
$ \mbox{MC}_{8}(-0.077) $	1.21e-13	3.32e-13	0.0032$ ^* $	0.0051$ ^* $	0	0.01	-0.01
$ \mbox{MC}_{8}(-0.073) $	6.93e-14	1.46e-13$ ^* $	5.55e-04$ ^* $	0.0139	-0.02	0.01	-0.03
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	3.91e-14	0.0142	4.80e-14	0.0167	-0.02	0.01	-0.03
$ \mbox{EC}_{10}(0.04) $	3.73e-14	0.0114	9.41e-14$ ^* $	0.0030$ ^* $	0	0.01	-0.01
$ \mbox{EC}_{10}(0.20) $	5.15e-14$ ^* $	1.82e-04$ ^* $	5.51e-14	0.0627	0.08	0.02	0.06
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	4.62e-14	5.68e-14	0.0358	0.0756	-0.10	0	-0.10
$ \mbox{MC}_{10}(0.19) $	4.26e-14	5.33e-14	0.0359$ ^* $	0.0051$ ^* $	0	0.01	-0.01
Narrow box	1.28e-13	0.0117	7.0014	0.0742	0.10	0.02	0.08
Multisymplectic	6.04e-14	0.0058	6.8991	0.2279	-0.31	-0.04	-0.27

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Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Sol. Err.	$ \text{Err}_{\phi_1} $	$ \text{Err}_{\phi_2} $	$ \text{Err}_{\phi} $
$ \mbox{EC}_{8}(0) $	1.74e-13	0.0036	5.13e-13	0.3701	-0.51	-0.06	-0.45
$ \mbox{EC}_{8}(1) $	1.33e-13	0.0732	8.01e-13$ ^* $	0.0085$ ^* $	0	-0.01	0.01
$ \mbox{EC}_{8}(-0.05) $	6.22e-14$ ^* $	1.81e-04$ ^* $	4.65e-13	0.3857	-0.53	-0.07	-0.46
$ \mbox{MC}_{8}(0) $	2.13e-13	3.69e-13	0.0632	0.2396	-0.32	-0.04	-0.28
$ \mbox{MC}_{8}(-0.077) $	1.21e-13	3.32e-13	0.0032$ ^* $	0.0051$ ^* $	0	0.01	-0.01
$ \mbox{MC}_{8}(-0.073) $	6.93e-14	1.46e-13$ ^* $	5.55e-04$ ^* $	0.0139	-0.02	0.01	-0.03
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	3.91e-14	0.0142	4.80e-14	0.0167	-0.02	0.01	-0.03
$ \mbox{EC}_{10}(0.04) $	3.73e-14	0.0114	9.41e-14$ ^* $	0.0030$ ^* $	0	0.01	-0.01
$ \mbox{EC}_{10}(0.20) $	5.15e-14$ ^* $	1.82e-04$ ^* $	5.51e-14	0.0627	0.08	0.02	0.06
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	4.62e-14	5.68e-14	0.0358	0.0756	-0.10	0	-0.10
$ \mbox{MC}_{10}(0.19) $	4.26e-14	5.33e-14	0.0359$ ^* $	0.0051$ ^* $	0	0.01	-0.01
Narrow box	1.28e-13	0.0117	7.0014	0.0742	0.10	0.02	0.08
Multisymplectic	6.04e-14	0.0058	6.8991	0.2279	-0.31	-0.04	-0.27

Table 2. Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $ \Delta x = 0.2 $, $ \Delta t = 0.05 $. An asterisk denotes the error that is minimized

Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Sol. Err.	$ \text{Err}_{\phi_1} $	$ \text{Err}_{\phi_2} $	$ \text{Err}_{\phi} $
$ \mbox{EC}_{8}(0) $	4.62e-14	0.0155	6.93e-14	0.9599	-1.84	-0.26	-1.58
$ \mbox{EC}_{8}(0.97) $	3.55e-14	0.2754	1.14e-13$ ^* $	0.0358$ ^* $	0	-0.03	0.03
$ \mbox{EC}_{8}(-0.06) $	4.26e-14$ ^* $	5.19e-04$ ^* $	1.15e-13	0.9798	-1.93	-0.26	-1.67
$ \mbox{MC}_{8}(0) $	4.44e-14	9.41e-14	0.2363	0.7553	-1.21	-0.15	-1.06
$ \mbox{MC}_{8}(-0.079) $	6.57e-14	1.42e-13	0.0138$ ^* $	0.0215$ ^* $	0	0.05	-0.05
$ \mbox{MC}_{8}(-0.075) $	4.09e-14	7.11e-14$ ^* $	0.0021$ ^* $	0.0567	-0.06	0.04	-0.10
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	2.13e-14	0.0574	3.73e-14	0.0725	-0.1	0.02	-0.12
$ \mbox{EC}_{10}(0.05) $	2.66e-14	0.0438	4.09e-14$ ^* $	0.0116$ ^* $	0	0.03	-0.03
$ \mbox{EC}_{10}(0.21) $	2.13e-14$ ^* $	6.74e-04$ ^* $	4.97e-14	0.2571	0.35	0.08	0.27
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	1.95e-14	4.80e-14	0.1461	0.2959	-0.40	-0.01	-0.39
$ \mbox{MC}_{10}(0.19) $	2.31e-14	2.49e-14	0.1477$ ^* $	0.0205$ ^* $	0	0.08	-0.08
Narrow box	4.97e-14	0.0459	6.8421	0.3054	0.40	0.09	0.31
Multisymplectic	2.66e-14	0.0228	6.4635	0.7278	-1.15	-0.17	-0.98

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Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Sol. Err.	$ \text{Err}_{\phi_1} $	$ \text{Err}_{\phi_2} $	$ \text{Err}_{\phi} $
$ \mbox{EC}_{8}(0) $	4.62e-14	0.0155	6.93e-14	0.9599	-1.84	-0.26	-1.58
$ \mbox{EC}_{8}(0.97) $	3.55e-14	0.2754	1.14e-13$ ^* $	0.0358$ ^* $	0	-0.03	0.03
$ \mbox{EC}_{8}(-0.06) $	4.26e-14$ ^* $	5.19e-04$ ^* $	1.15e-13	0.9798	-1.93	-0.26	-1.67
$ \mbox{MC}_{8}(0) $	4.44e-14	9.41e-14	0.2363	0.7553	-1.21	-0.15	-1.06
$ \mbox{MC}_{8}(-0.079) $	6.57e-14	1.42e-13	0.0138$ ^* $	0.0215$ ^* $	0	0.05	-0.05
$ \mbox{MC}_{8}(-0.075) $	4.09e-14	7.11e-14$ ^* $	0.0021$ ^* $	0.0567	-0.06	0.04	-0.10
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	2.13e-14	0.0574	3.73e-14	0.0725	-0.1	0.02	-0.12
$ \mbox{EC}_{10}(0.05) $	2.66e-14	0.0438	4.09e-14$ ^* $	0.0116$ ^* $	0	0.03	-0.03
$ \mbox{EC}_{10}(0.21) $	2.13e-14$ ^* $	6.74e-04$ ^* $	4.97e-14	0.2571	0.35	0.08	0.27
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	1.95e-14	4.80e-14	0.1461	0.2959	-0.40	-0.01	-0.39
$ \mbox{MC}_{10}(0.19) $	2.31e-14	2.49e-14	0.1477$ ^* $	0.0205$ ^* $	0	0.08	-0.08
Narrow box	4.97e-14	0.0459	6.8421	0.3054	0.40	0.09	0.31
Multisymplectic	2.66e-14	0.0228	6.4635	0.7278	-1.15	-0.17	-0.98

Table 3. Errors in conservation laws and solution for the breather problem, setting $ \Delta x = 0.02 $, $ \Delta t = 0.002 $. An asterisk denotes the error that is minimized

Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Solution error
$ \mbox{EC}_{8}(0) $	6.76e-13	0.1091	1.33e-10	0.9099
$ \mbox{EC}_{8}(2.22) $	2.97e-13	0.3979	1.55e-10$ ^* $	0.0144$ ^* $
$ \mbox{EC}_{8}(0.49) $	9.59e-13$ ^* $	0.0079$ ^* $	1.05e-10	0.7442
$ \mbox{MC}_{8}(0) $	6.25e-13	5.31e-12	7.534	0.7666
$ \mbox{MC}_{8}(-0.165) $	3.03e-13	2.74e-12	2.3599$ ^* $	0.0497$ ^* $
$ \mbox{MC}_{8}(-0.128) $	7.24e-13	7.60e-12$ ^* $	0.1728$ ^* $	0.1931
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	3.28e-14	0.1765	1.24e-11	0.4042
$ \mbox{EC}_{10}(0.92) $	3.53e-14	0.0296	2.63e-11$ ^* $	0.0295$ ^* $
$ \mbox{EC}_{10}(0.78) $	5.96e-14$ ^* $	0.0095$ ^* $	1.35e-11	0.0708
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	1.14e-13	9.24e-13	4.3586	0.5040
$ \mbox{MC}_{10}(1.15) $	5.34e-14	7.03e-13	4.8298$ ^* $	0.0219$ ^* $
Narrow box	1.00e-12	0.0382	566.37	0.3477
Multisymplectic	2.60e-13	0.0184	539.40	0.7994

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Method	$ \text{Err}_1 $	$ \text{Err}_2 $	$ \text{Err}_3 $	Solution error
$ \mbox{EC}_{8}(0) $	6.76e-13	0.1091	1.33e-10	0.9099
$ \mbox{EC}_{8}(2.22) $	2.97e-13	0.3979	1.55e-10$ ^* $	0.0144$ ^* $
$ \mbox{EC}_{8}(0.49) $	9.59e-13$ ^* $	0.0079$ ^* $	1.05e-10	0.7442
$ \mbox{MC}_{8}(0) $	6.25e-13	5.31e-12	7.534	0.7666
$ \mbox{MC}_{8}(-0.165) $	3.03e-13	2.74e-12	2.3599$ ^* $	0.0497$ ^* $
$ \mbox{MC}_{8}(-0.128) $	7.24e-13	7.60e-12$ ^* $	0.1728$ ^* $	0.1931
$ \mbox{AVF}_\text{EC} $; $ \mbox{EC}_{10}(0) $	3.28e-14	0.1765	1.24e-11	0.4042
$ \mbox{EC}_{10}(0.92) $	3.53e-14	0.0296	2.63e-11$ ^* $	0.0295$ ^* $
$ \mbox{EC}_{10}(0.78) $	5.96e-14$ ^* $	0.0095$ ^* $	1.35e-11	0.0708
$ \mbox{AVF}_\text{MC} $; $ \mbox{MC}_{10}(0) $	1.14e-13	9.24e-13	4.3586	0.5040
$ \mbox{MC}_{10}(1.15) $	5.34e-14	7.03e-13	4.8298$ ^* $	0.0219$ ^* $
Narrow box	1.00e-12	0.0382	566.37	0.3477
Multisymplectic	2.60e-13	0.0184	539.40	0.7994

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Locally conservative finite difference schemes for the modified KdV equation

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