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Locally conservative finite difference schemes for the modified KdV equation
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK |
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.
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. |




Method | |
Sol. Err. | |||||
1.74e-13 | 0.0036 | 5.13e-13 | 0.3701 | -0.51 | -0.06 | -0.45 | |
1.33e-13 | 0.0732 | 8.01e-13 |
0.0085 |
0 | -0.01 | 0.01 | |
6.22e-14 |
1.81e-04 |
4.65e-13 | 0.3857 | -0.53 | -0.07 | -0.46 | |
2.13e-13 | 3.69e-13 | 0.0632 | 0.2396 | -0.32 | -0.04 | -0.28 | |
1.21e-13 | 3.32e-13 | 0.0032 |
0.0051 |
0 | 0.01 | -0.01 | |
6.93e-14 | 1.46e-13 |
5.55e-04 |
0.0139 | -0.02 | 0.01 | -0.03 | |
3.91e-14 | 0.0142 | 4.80e-14 | 0.0167 | -0.02 | 0.01 | -0.03 | |
3.73e-14 | 0.0114 | 9.41e-14 |
0.0030 |
0 | 0.01 | -0.01 | |
5.15e-14 |
1.82e-04 |
5.51e-14 | 0.0627 | 0.08 | 0.02 | 0.06 | |
4.62e-14 | 5.68e-14 | 0.0358 | 0.0756 | -0.10 | 0 | -0.10 | |
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 | |
Sol. Err. | |||||
1.74e-13 | 0.0036 | 5.13e-13 | 0.3701 | -0.51 | -0.06 | -0.45 | |
1.33e-13 | 0.0732 | 8.01e-13 |
0.0085 |
0 | -0.01 | 0.01 | |
6.22e-14 |
1.81e-04 |
4.65e-13 | 0.3857 | -0.53 | -0.07 | -0.46 | |
2.13e-13 | 3.69e-13 | 0.0632 | 0.2396 | -0.32 | -0.04 | -0.28 | |
1.21e-13 | 3.32e-13 | 0.0032 |
0.0051 |
0 | 0.01 | -0.01 | |
6.93e-14 | 1.46e-13 |
5.55e-04 |
0.0139 | -0.02 | 0.01 | -0.03 | |
3.91e-14 | 0.0142 | 4.80e-14 | 0.0167 | -0.02 | 0.01 | -0.03 | |
3.73e-14 | 0.0114 | 9.41e-14 |
0.0030 |
0 | 0.01 | -0.01 | |
5.15e-14 |
1.82e-04 |
5.51e-14 | 0.0627 | 0.08 | 0.02 | 0.06 | |
4.62e-14 | 5.68e-14 | 0.0358 | 0.0756 | -0.10 | 0 | -0.10 | |
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 | |
Sol. Err. | |||||
4.62e-14 | 0.0155 | 6.93e-14 | 0.9599 | -1.84 | -0.26 | -1.58 | |
3.55e-14 | 0.2754 | 1.14e-13 |
0.0358 |
0 | -0.03 | 0.03 | |
4.26e-14 |
5.19e-04 |
1.15e-13 | 0.9798 | -1.93 | -0.26 | -1.67 | |
4.44e-14 | 9.41e-14 | 0.2363 | 0.7553 | -1.21 | -0.15 | -1.06 | |
6.57e-14 | 1.42e-13 | 0.0138 |
0.0215 |
0 | 0.05 | -0.05 | |
4.09e-14 | 7.11e-14 |
0.0021 |
0.0567 | -0.06 | 0.04 | -0.10 | |
2.13e-14 | 0.0574 | 3.73e-14 | 0.0725 | -0.1 | 0.02 | -0.12 | |
2.66e-14 | 0.0438 | 4.09e-14 |
0.0116 |
0 | 0.03 | -0.03 | |
2.13e-14 |
6.74e-04 |
4.97e-14 | 0.2571 | 0.35 | 0.08 | 0.27 | |
1.95e-14 | 4.80e-14 | 0.1461 | 0.2959 | -0.40 | -0.01 | -0.39 | |
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 | |
Sol. Err. | |||||
4.62e-14 | 0.0155 | 6.93e-14 | 0.9599 | -1.84 | -0.26 | -1.58 | |
3.55e-14 | 0.2754 | 1.14e-13 |
0.0358 |
0 | -0.03 | 0.03 | |
4.26e-14 |
5.19e-04 |
1.15e-13 | 0.9798 | -1.93 | -0.26 | -1.67 | |
4.44e-14 | 9.41e-14 | 0.2363 | 0.7553 | -1.21 | -0.15 | -1.06 | |
6.57e-14 | 1.42e-13 | 0.0138 |
0.0215 |
0 | 0.05 | -0.05 | |
4.09e-14 | 7.11e-14 |
0.0021 |
0.0567 | -0.06 | 0.04 | -0.10 | |
2.13e-14 | 0.0574 | 3.73e-14 | 0.0725 | -0.1 | 0.02 | -0.12 | |
2.66e-14 | 0.0438 | 4.09e-14 |
0.0116 |
0 | 0.03 | -0.03 | |
2.13e-14 |
6.74e-04 |
4.97e-14 | 0.2571 | 0.35 | 0.08 | 0.27 | |
1.95e-14 | 4.80e-14 | 0.1461 | 0.2959 | -0.40 | -0.01 | -0.39 | |
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 | |
Solution error | ||
6.76e-13 | 0.1091 | 1.33e-10 | 0.9099 | |
2.97e-13 | 0.3979 | 1.55e-10 |
0.0144 |
|
9.59e-13 |
0.0079 |
1.05e-10 | 0.7442 | |
6.25e-13 | 5.31e-12 | 7.534 | 0.7666 | |
3.03e-13 | 2.74e-12 | 2.3599 |
0.0497 |
|
7.24e-13 | 7.60e-12 |
0.1728 |
0.1931 | |
3.28e-14 | 0.1765 | 1.24e-11 | 0.4042 | |
3.53e-14 | 0.0296 | 2.63e-11 |
0.0295 |
|
5.96e-14 |
0.0095 |
1.35e-11 | 0.0708 | |
1.14e-13 | 9.24e-13 | 4.3586 | 0.5040 | |
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 |
Method | |
Solution error | ||
6.76e-13 | 0.1091 | 1.33e-10 | 0.9099 | |
2.97e-13 | 0.3979 | 1.55e-10 |
0.0144 |
|
9.59e-13 |
0.0079 |
1.05e-10 | 0.7442 | |
6.25e-13 | 5.31e-12 | 7.534 | 0.7666 | |
3.03e-13 | 2.74e-12 | 2.3599 |
0.0497 |
|
7.24e-13 | 7.60e-12 |
0.1728 |
0.1931 | |
3.28e-14 | 0.1765 | 1.24e-11 | 0.4042 | |
3.53e-14 | 0.0296 | 2.63e-11 |
0.0295 |
|
5.96e-14 |
0.0095 |
1.35e-11 | 0.0708 | |
1.14e-13 | 9.24e-13 | 4.3586 | 0.5040 | |
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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