American Institute of Mathematical Sciences

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May  2022, 27(5): 2441-2453. doi: 10.3934/dcdsb.2021139

Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

Jiaxiang Cai 1,2,, , Juan Chen 2, and  Min Chen 1,

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

* Corresponding author: cjx1981@hytc.edu.cn (J. Cai)

Received  November 2020 Revised  March 2021 Published  May 2022 Early access  May 2021

Fund Project: The first author is supported by the Natural Science Foundation of Jiangsu Province of China grant BK20181482, China Postdoctoral Science Foundation through grant 2020M671532, Jiangsu Province Postdoctoral Science Foundation through grant 2020Z147 and Qing Lan Project of Jiangsu Province of 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.

Keywords: Kadomtsev-Petviashvili equation, Structure-preserving algorithm, Hamiltonian PDE, Kahan's method, energy-preserving method.
Mathematics Subject Classification: 65P10, 65N35, 65N06.
Citation: Jiaxiang Cai, Juan Chen, Min Chen. Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2441-2453. doi: 10.3934/dcdsb.2021139
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.

Figure 1.  Left: the errors in solution (Solid line: $ {\rm e}_{\infty} $; Dashed line: $ {\rm e}_2 $); Right: waveform at time $ t = 12 $
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Figure 2.  The lump type solitary wave at time $ t = 0 $ (top), $ t = 0.5 $ (the second row), $ t = 1 $ (the third row) and $ t = 5 $ (bottom), respectively. Left: the profiles of solution; Middle: contours of numerical solution; Right: contours of exact solution
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Figure 3.  The lump type solitary wave at time $ t = 20 $. Left: the profile of solution; Middle: contours of numerical solution; Right: contours of exact solution
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Figure 4.  The profiles of interaction of the two lump waves at different times and the variation of the global energy (the last graph): $ \Delta t = 1 $e-2, $ \Delta x = \Delta y = 2 $e-1
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Figure 5.  The motion of single soliton for the KPII equation. Top: $ t = 0 $; Bottom: $ t = 8 $
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Table 1.  Numerical results for the KPI equation obtained by the present scheme: $ \Omega = [0,40]\times[0,2] $, $ r = (\Delta t,\Delta x,\Delta y) = (0.1,0.2,0.2) $ and $ t = 1 $
Mesh $r$ $r/2$ $r/2^2$ $r/2^3$ $r/2^4$
${\rm e}_{\infty}$ 4.4921e-2 1.1041e-2 2.8764e-3 7.1999e-4 1.6418e-4
Rate - 2.0248 1.9405 1.9982 2.1327
${\rm e}_2$ 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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Mesh $r$ $r/2$ $r/2^2$ $r/2^3$ $r/2^4$
${\rm e}_{\infty}$ 4.4921e-2 1.1041e-2 2.8764e-3 7.1999e-4 1.6418e-4
Rate - 2.0248 1.9405 1.9982 2.1327
${\rm e}_2$ 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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