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

Locally conservative finite difference schemes for the modified KdV equation

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.

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. AncoM. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[5]

L. BarlettiL. BrugnanoG. 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.  Google Scholar

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[7]

T. J. BridgesP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[12]

E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealB. 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.  Google Scholar

[13]

E. CelledoniR. I. McLachlanB. 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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. Gonzales, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467.  doi: 10.1007/BF02440162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. A. Kuperschmidt, Discrete Lax equations and differential-difference calculus, Astérisque, (1985), 212pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

F. McDonaldR. I. McLachlanB. 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.  Google Scholar

[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.  Google Scholar

[36]

R. I. McLachlanG. 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.  Google Scholar

[37]

R. MiuraC. 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.  Google Scholar

[38]

M. OliverM. 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.  Google Scholar

[39]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Mathematical Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. C. AncoM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

L. BarlettiL. BrugnanoG. 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.  Google Scholar

[6]

T. J. Bridges, Multisymplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc., 121 (1997), 147-190.  doi: 10.1017/S0305004196001429.  Google Scholar

[7]

T. J. BridgesP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. CelledoniV. GrimmR. I. McLachlanD. I. McLarenD. O'NealB. 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.  Google Scholar

[13]

E. CelledoniR. I. McLachlanB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

O. Gonzales, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6 (1996), 449-467.  doi: 10.1007/BF02440162.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. A. Kuperschmidt, Discrete Lax equations and differential-difference calculus, Astérisque, (1985), 212pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

F. McDonaldR. I. McLachlanB. 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.  Google Scholar

[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.  Google Scholar

[36]

R. I. McLachlanG. 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.  Google Scholar

[37]

R. MiuraC. 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.  Google Scholar

[38]

M. OliverM. 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.  Google Scholar

[39]

P. J. Olver, Evolution equations possessing infinitely many symmetries, J. Mathematical Phys., 18 (1977), 1212-1215.  doi: 10.1063/1.523393.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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)
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)
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)
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
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
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 539.400.7994
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 539.400.7994
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