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Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation
1. | School of Geography, Nanjing Normal University, Nanjing, Jiangsu, 210023, China |
2. | School of Mathematics and Statistics, Huaiyin Normal University, Huaian, Jiangsu 223300, China |
3. | Department of Basic Education, Jiangsu Vocational College of Finance & Economics, Huaian, Jiangsu, 223003, China |
A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.
References:
[1] |
K. M. Berger and P. A. Milewski,
The Generation and evolution of lump solitary waves in surface-tension-dominated flows, SIAM J. Appl. Math., 61 (2000), 731-750.
doi: 10.1137/S0036139999356971. |
[2] |
A. G. Bratsos and E. H. Twizell,
An explicit finite-difference scheme for the solution of the Kadomtsev-Petviashvili equation, Inter. J. Comput. Math., 68 (1998), 175-187.
doi: 10.1080/00207169808804685. |
[3] |
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. |
[4] |
J. Cai, Y. Wang and C. Jiang,
Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs, Comput. Phys. Commun., 235 (2019), 210-220.
doi: 10.1016/j.cpc.2018.08.015. |
[5] |
J. Cai and S. Jie, Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 401 (2020), 108975, 17pp.
doi: 10.1016/j.jcp.2019.108975. |
[6] |
E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12pp.
doi: 10.1088/1751-8113/46/2/025201. |
[7] |
E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20pp.
doi: 10.1088/1751-8113/47/36/365202. |
[8] |
E. Celledoni, D. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: the preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9pp.
doi: 10.1088/1751-8121/aafb1e. |
[9] |
E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Discretization of polynomial vector fields by polarization, Proc. R. Soc. A: Math. Phys., 471 (2015), 20150390, 10pp.
doi: 10.1098/rspa.2015.0390. |
[10] |
S. Eidnes and L. Li, Linearly implicit local and global energy-preserving methods for Hamiltonian PDEs, SIAM J. Sci. Comput., 42 (2020), A2865–A2888, arXiv: 1907.02122v1.
doi: 10.1137/19M1272688. |
[11] |
L. Einkemmer and A. Ostermann,
A split step Fourier/discontinuous Galerkin scheme for the Kadomtsev-Petviashvili equation, Appl. Math. Comput., 334 (2018), 311-325.
doi: 10.1016/j.amc.2018.04.013. |
[12] |
L. Einkemmer and A. Ostermann,
A splitting approach for the Kadomtsev-Petviashvili equation, J. Comput. Phys., 299 (2015), 716-730.
doi: 10.1016/j.jcp.2015.07.024. |
[13] |
B. F. Feng and T. Mitsui,
A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. Comput. Appl. Math., 90 (1998), 95-116.
doi: 10.1016/S0377-0427(98)00006-5. |
[14] |
B. Fornberg and G. B. Whitham,
A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Soc. London, 289 (1978), 373-404.
doi: 10.1098/rsta.1978.0064. |
[15] |
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. |
[16] |
Q. Hong, Y. S. Wang and Q. K. 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. |
[17] |
E. Infeld, A. Senatorski and A. Skorupski,
Numerical simulations of Kadomtsev-Petviashvili soliton interactions, Phys. Rev. E, 51 (1995), 3183-3191.
doi: 10.1103/PhysRevE.51.3183. |
[18] |
B. Jiang, Y. Wang and J. Cai, A new multisymplectic scheme for generalized Kadomtsev-Petviashvili equation, J. Math. Phys., 47 (2006), 083503, 14pp.
doi: 10.1063/1.2234261. |
[19] |
B. B. Kadomtsev and V. I. Petviashvili,
On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl, 15 (1970), 539-541.
|
[20] |
M. Kumar, A. K. Tiwari and R. Kumar,
Some more solutions of Kadomtsev-Petviashvili equation, Comput. Math. Appl., 74 (2017), 2599-2607.
doi: 10.1016/j.camwa.2017.07.034. |
[21] |
C. Katsis and T. R. Akylas,
Solitary internal wave in a rotating channel: A numerical study, Phys. Fluid, 30 (1987), 297-301.
doi: 10.1063/1.866377. |
[22] |
W. Kahan,
Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, 1 (1993), 1-15.
|
[23] |
W. Kahan and R. C. Li,
Unconventional schemes for a class of ordinary differential equations–with applications to the Korteweg-de Vries equation, J. Comput. Phys., 134 (1997), 316-331.
doi: 10.1006/jcph.1997.5710. |
[24] |
C. Klein and K. Roidot,
Fourth Order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356.
doi: 10.1137/100816663. |
[25] |
C. Klein, C. Sparber and P. Markowich,
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonlinear Sci., 17 (2007), 429-470.
doi: 10.1007/s00332-007-9001-y. |
[26] |
G. A. Latham,
Solutions of the KP equation associated to rank-three commuting differential operators over a singular elliptic curve, Physica D, 41 (1990), 55-66.
doi: 10.1016/0167-2789(90)90027-M. |
[27] |
Y. W. Li and X. Wu,
General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 301 (2015), 141-166.
doi: 10.1016/j.jcp.2015.08.023. |
[28] |
T. Liu and M. Qin,
Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation, J. Math. Phys., 43 (2002), 4060-4077.
doi: 10.1063/1.1487444. |
[29] |
J. E. Marsden, G. P. Patrick and S. Shkoller,
Multi-symplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.
doi: 10.1007/s002200050505. |
[30] |
A. A. Minzoni and N. F. Smyth,
Evolution of lump solutions for the KP equation, Wave Motion, 24 (1996), 291-305.
doi: 10.1016/S0165-2125(96)00023-6. |
[31] |
B. E. Moore and S. Reich,
Backward error analysis for multi-symplectic integration methods, Numer. Math., 95 (2003), 625-652.
doi: 10.1007/s00211-003-0458-9. |
[32] |
G. Pitton, Numerical study of dispersive shock waves in the KPI equation, Doctoral Thesis, Math.Sissa.It, (2018). |
[33] |
S. Reich,
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), 473-499.
doi: 10.1006/jcph.1999.6372. |
[34] |
A. M. Wazwaz,
A computational approach to soliton solutions of the Kadomtsev-Petviashvili equation, Appl. Math. Comput., 123 (2001), 205-217.
doi: 10.1016/S0096-3003(00)00065-5. |
[35] |
Y. S. Wang, B. Wang and M. Z. Qin,
Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A, 51 (2008), 2115-2136.
doi: 10.1007/s11425-008-0046-7. |
show all references
References:
[1] |
K. M. Berger and P. A. Milewski,
The Generation and evolution of lump solitary waves in surface-tension-dominated flows, SIAM J. Appl. Math., 61 (2000), 731-750.
doi: 10.1137/S0036139999356971. |
[2] |
A. G. Bratsos and E. H. Twizell,
An explicit finite-difference scheme for the solution of the Kadomtsev-Petviashvili equation, Inter. J. Comput. Math., 68 (1998), 175-187.
doi: 10.1080/00207169808804685. |
[3] |
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. |
[4] |
J. Cai, Y. Wang and C. Jiang,
Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs, Comput. Phys. Commun., 235 (2019), 210-220.
doi: 10.1016/j.cpc.2018.08.015. |
[5] |
J. Cai and S. Jie, Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 401 (2020), 108975, 17pp.
doi: 10.1016/j.jcp.2019.108975. |
[6] |
E. Celledoni, R. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric properties of Kahan's method, J. Phys. A, 46 (2013), 025201, 12pp.
doi: 10.1088/1751-8113/46/2/025201. |
[7] |
E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202, 20pp.
doi: 10.1088/1751-8113/47/36/365202. |
[8] |
E. Celledoni, D. I. McLachlan, B. Owren and G. R. W. Quispel, Geometric and integrability properties of Kahan's method: the preservation of certain quadratic integrals, J. Phys. A, 52 (2019), 065201, 9pp.
doi: 10.1088/1751-8121/aafb1e. |
[9] |
E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Discretization of polynomial vector fields by polarization, Proc. R. Soc. A: Math. Phys., 471 (2015), 20150390, 10pp.
doi: 10.1098/rspa.2015.0390. |
[10] |
S. Eidnes and L. Li, Linearly implicit local and global energy-preserving methods for Hamiltonian PDEs, SIAM J. Sci. Comput., 42 (2020), A2865–A2888, arXiv: 1907.02122v1.
doi: 10.1137/19M1272688. |
[11] |
L. Einkemmer and A. Ostermann,
A split step Fourier/discontinuous Galerkin scheme for the Kadomtsev-Petviashvili equation, Appl. Math. Comput., 334 (2018), 311-325.
doi: 10.1016/j.amc.2018.04.013. |
[12] |
L. Einkemmer and A. Ostermann,
A splitting approach for the Kadomtsev-Petviashvili equation, J. Comput. Phys., 299 (2015), 716-730.
doi: 10.1016/j.jcp.2015.07.024. |
[13] |
B. F. Feng and T. Mitsui,
A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations, J. Comput. Appl. Math., 90 (1998), 95-116.
doi: 10.1016/S0377-0427(98)00006-5. |
[14] |
B. Fornberg and G. B. Whitham,
A numerical and theoretical study of certain nonlinear wave phenomena, Phil. Trans. Roy. Soc. London, 289 (1978), 373-404.
doi: 10.1098/rsta.1978.0064. |
[15] |
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. |
[16] |
Q. Hong, Y. S. Wang and Q. K. 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. |
[17] |
E. Infeld, A. Senatorski and A. Skorupski,
Numerical simulations of Kadomtsev-Petviashvili soliton interactions, Phys. Rev. E, 51 (1995), 3183-3191.
doi: 10.1103/PhysRevE.51.3183. |
[18] |
B. Jiang, Y. Wang and J. Cai, A new multisymplectic scheme for generalized Kadomtsev-Petviashvili equation, J. Math. Phys., 47 (2006), 083503, 14pp.
doi: 10.1063/1.2234261. |
[19] |
B. B. Kadomtsev and V. I. Petviashvili,
On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl, 15 (1970), 539-541.
|
[20] |
M. Kumar, A. K. Tiwari and R. Kumar,
Some more solutions of Kadomtsev-Petviashvili equation, Comput. Math. Appl., 74 (2017), 2599-2607.
doi: 10.1016/j.camwa.2017.07.034. |
[21] |
C. Katsis and T. R. Akylas,
Solitary internal wave in a rotating channel: A numerical study, Phys. Fluid, 30 (1987), 297-301.
doi: 10.1063/1.866377. |
[22] |
W. Kahan,
Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, 1 (1993), 1-15.
|
[23] |
W. Kahan and R. C. Li,
Unconventional schemes for a class of ordinary differential equations–with applications to the Korteweg-de Vries equation, J. Comput. Phys., 134 (1997), 316-331.
doi: 10.1006/jcph.1997.5710. |
[24] |
C. Klein and K. Roidot,
Fourth Order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations, SIAM J. Sci. Comput., 33 (2011), 3333-3356.
doi: 10.1137/100816663. |
[25] |
C. Klein, C. Sparber and P. Markowich,
Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation, J. Nonlinear Sci., 17 (2007), 429-470.
doi: 10.1007/s00332-007-9001-y. |
[26] |
G. A. Latham,
Solutions of the KP equation associated to rank-three commuting differential operators over a singular elliptic curve, Physica D, 41 (1990), 55-66.
doi: 10.1016/0167-2789(90)90027-M. |
[27] |
Y. W. Li and X. Wu,
General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 301 (2015), 141-166.
doi: 10.1016/j.jcp.2015.08.023. |
[28] |
T. Liu and M. Qin,
Multisymplectic geometry and multisymplectic Preissman scheme for the KP equation, J. Math. Phys., 43 (2002), 4060-4077.
doi: 10.1063/1.1487444. |
[29] |
J. E. Marsden, G. P. Patrick and S. Shkoller,
Multi-symplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395.
doi: 10.1007/s002200050505. |
[30] |
A. A. Minzoni and N. F. Smyth,
Evolution of lump solutions for the KP equation, Wave Motion, 24 (1996), 291-305.
doi: 10.1016/S0165-2125(96)00023-6. |
[31] |
B. E. Moore and S. Reich,
Backward error analysis for multi-symplectic integration methods, Numer. Math., 95 (2003), 625-652.
doi: 10.1007/s00211-003-0458-9. |
[32] |
G. Pitton, Numerical study of dispersive shock waves in the KPI equation, Doctoral Thesis, Math.Sissa.It, (2018). |
[33] |
S. Reich,
Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), 473-499.
doi: 10.1006/jcph.1999.6372. |
[34] |
A. M. Wazwaz,
A computational approach to soliton solutions of the Kadomtsev-Petviashvili equation, Appl. Math. Comput., 123 (2001), 205-217.
doi: 10.1016/S0096-3003(00)00065-5. |
[35] |
Y. S. Wang, B. Wang and M. Z. Qin,
Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A, 51 (2008), 2115-2136.
doi: 10.1007/s11425-008-0046-7. |




Mesh | |||||
4.4921e-2 | 1.1041e-2 | 2.8764e-3 | 7.1999e-4 | 1.6418e-4 | |
Rate | - | 2.0248 | 1.9405 | 1.9982 | 2.1327 |
5.8460e-2 | 1.3052e-2 | 3.2214e-3 | 8.0842e-4 | 1.9908e-4 | |
Rate | - | 2.1632 | 2.0185 | 1.9945 | 2.0218 |
Mesh | |||||
4.4921e-2 | 1.1041e-2 | 2.8764e-3 | 7.1999e-4 | 1.6418e-4 | |
Rate | - | 2.0248 | 1.9405 | 1.9982 | 2.1327 |
5.8460e-2 | 1.3052e-2 | 3.2214e-3 | 8.0842e-4 | 1.9908e-4 | |
Rate | - | 2.1632 | 2.0185 | 1.9945 | 2.0218 |
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