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doi: 10.3934/dcdsb.2021139
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Kahan, Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, 1 (1993), 1-15.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

G. Pitton, Numerical study of dispersive shock waves in the KPI equation, Doctoral Thesis, Math.Sissa.It, (2018). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

W. Kahan, Unconventional numerical methods for trajectory calculations, Unpublished Lecture Notes, 1 (1993), 1-15.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

G. Pitton, Numerical study of dispersive shock waves in the KPI equation, Doctoral Thesis, Math.Sissa.It, (2018). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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