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Linearly implicit local energy-preserving algorithm for a class of multi-symplectic Hamiltonian PDEs

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Abstract

We give a systematic, linearly implicit, local energy-preserving method for the general multi-symplectic Hamiltonian system with cubic invariants by combining Kahan’s discretization in time and Euler-box discretization in space. The new method is applied to some classical multi-symplectic Hamiltonian partial differential equations to test its effectiveness, efficiency, solution accuracy and energy preservation.

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References

  • Ascher UM, McLachlan RI (2005) On symplectic and multisymplectic schemes for the KdV equation. J Sci Comput 25:83–104

    Article  MathSciNet  Google Scholar 

  • Bridges TJ, Reich S (2001a) Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys Lett A 284:184–193

    Article  MathSciNet  Google Scholar 

  • Bridges TJ, Reich S (2001b) Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations. Phys D 152:491–504

    Article  MathSciNet  Google Scholar 

  • Cai J, Shen J (2020) Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs. J Comput Phys 401:108975

    Article  MathSciNet  Google Scholar 

  • Cai J, Wang Y, Liang H (2013) Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system. J Comput Phys 239:30–50

    Article  MathSciNet  Google Scholar 

  • Cai J, Wang Y, Gong Y (2016) Numerical analysis of AVF methods for three-dimensional time-domain Maxwell’s equations. J Sci Comput 66:141–176

    Article  MathSciNet  Google Scholar 

  • Cai J, Bai C, Zhang H (2018) Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations. J Comput Phys 374:281–299

    Article  MathSciNet  Google Scholar 

  • Cai J, Wang Y, Jiang C (2019) Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs. Comput Phys Commun 235:210–220

    Article  MathSciNet  Google Scholar 

  • Camassa R, Holm D (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–1664

    Article  MathSciNet  Google Scholar 

  • Celledoni E, McLachlan RI, Owren B, Quispel GRW (2013) Geometric properties of Kahan’s method. J Phys A 46:025201

    Article  MathSciNet  Google Scholar 

  • Celledoni E, McLachlan RI, McLaren DI, Owren B, Quispel GRW (2014) Integrability properties of Kahan’s method. J Phys A 47:365202

    Article  MathSciNet  Google Scholar 

  • Celledoni E, McLachlan RI, McLaren DI, Owren B, Quispel GRW (2015) Discretization of polynomial vector fields by polarization. Proc R Soc A Math Phys 471:20150390

    MathSciNet  MATH  Google Scholar 

  • Celledoni E, McLachlan DI, Owren B, Quispel GRW (2019) Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals. J Phys A 52:065201

    Article  MathSciNet  Google Scholar 

  • Chen J, Qin M (2001) Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation. Electron Trans Numer Anal 12:193–204

    MathSciNet  MATH  Google Scholar 

  • Cohen D, Owren B, Raynaud X (2008) Multi-symplectic integration of the Camassa–Holm equation. J Comput Phys 227:5492–5512

    Article  MathSciNet  Google Scholar 

  • Dahlby M, Owren B (2011) A general framework for deriving integral preserving numerical methods for PDEs. SIAM J Sci Comput 33:2318–2340

    Article  MathSciNet  Google Scholar 

  • Eidnes S, Li L (2019) Linearly implicit local and global energy-preserving methods for Hamiltonian PDEs, arXiv: 1907.02122v1,

  • Gong Y, Wang Y (2016) An energy-preserving wavelet collocation method for general multi-symplectic formulations of Hamiltonian PDEs. Commun Comput Phys 20:1313–1339

    Article  MathSciNet  Google Scholar 

  • Gong Y, Cai J, Wang Y (2014) Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs. J Comput Phys 279:80–102

    Article  MathSciNet  Google Scholar 

  • Hong J, Jiang S, Li C (2009) Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations. J Comput Phys 228:3517–3532

    Article  MathSciNet  Google Scholar 

  • Hong Q, Gong Y, Lv Z (2019) Linearly and Hamiltonian-conserving Fourier pseudo-spectral schemes for the Camassa–Holm equation. Appl Math Comput 346:86–95

    MathSciNet  MATH  Google Scholar 

  • Kahan W (1993) Unconventional numerical methods for trajectory calculations, Unpublished lecture notes, 1 1-15

  • Kahan W, Li RC (1997) Unconventional schemes for a class of ordinary differential equations-with applications to the Korteweg–de Vries equation. J Comput Phys 134:316–331

    Article  MathSciNet  Google Scholar 

  • Kalisch H, Lenells J (2005) Numerical study of traveling-wave solutions for the Camassa–Holm equation. Chaos Solitons Frac 25:287–298

    Article  MathSciNet  Google Scholar 

  • Kalisch H, Raynaud X (2006) Convergence of a spectral projection of the Camassa-Holm equation, Numer. Meth. P. D. E. 22 1197-1215

  • Kong L, Hong J, Zhang J (2010) Splitting multisymplectic integrators for Maxwell’s equations. J Comput Phys 229:4259–4278

    Article  MathSciNet  Google Scholar 

  • Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monogr. Appl. Comput. Math. Cambridge University Press, Cambridge

  • Matsuo T, Furihata D (2001) Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations. J Comput Phys 171:425–447

    Article  MathSciNet  Google Scholar 

  • Mu Z, Gong Y, Cai W, Wang Y (2018) Efficient local energy dissipation preserving algorithms for the Cahn–Hilliard equation. J Comput Phys 374:654–667

    Article  MathSciNet  Google Scholar 

  • Quispel GRW, McLaren DI (2008) A new class of energy-preserving numerical integration methods. J Phys A 41(045206):1–7

    MathSciNet  MATH  Google Scholar 

  • Sun Y, Tse PSP (2011) Symplectic and multisymplectic numerical methods for Maxwell’s equations. J Comput Phys 230:2076–2094

    Article  MathSciNet  Google Scholar 

  • Wang Y, Wang B, Qin M (2008) Local structure-preserving algorithms for partial differential equations. Sci China Ser A Math 51:2115–2136

    Article  MathSciNet  Google Scholar 

  • Wang J, Wang Y, Liang D (2020) Construction of the local structure-preserving algorithms for the general multi-symplectic Hamiltonian system. Commun Comput Phys 27:828–860

    Article  MathSciNet  Google Scholar 

  • Xu Y, Shu CW (2008) A local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J Numer Anal 46:1998–2021

    Article  MathSciNet  Google Scholar 

  • Yang X (2016) Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J Comput Phys 327:294–316

    Article  MathSciNet  Google Scholar 

  • Zakharov V, Kuznetsov E (1974) Three-dimensional solitons. Sov Phys JETP 39:285–286

    Google Scholar 

  • Zhao P, Qin M (2000) Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation. J Phys A Math Gen 33:3613–3626

    Article  MathSciNet  Google Scholar 

  • Zhu H, Song S, Chen Y (2011a) Multi-symplectic wavelet collocation method for Maxwell’s equations. Adv Appl Math Mech 3:663–688

    Article  MathSciNet  Google Scholar 

  • Zhu H, Song S, Tang Y (2011b) Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation. Comput Phys Commun 182:616–627

    Article  Google Scholar 

Download references

Acknowledgements

The work is funded by Natural Science Foundation of Jiangsu Province of China (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.

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Authors and Affiliations

  1. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, Jiangsu, China

    Jiaxiang Cai & Bangyu Shen

  2. School of Geography, Nanjing Normal University, Nanjing, 210023, Jiangsu, China

    Jiaxiang Cai

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  1. Jiaxiang Cai
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  2. Bangyu Shen
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Correspondence to Jiaxiang Cai.

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Communicated by Zhaosheng Feng.

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Cai, J., Shen, B. Linearly implicit local energy-preserving algorithm for a class of multi-symplectic Hamiltonian PDEs. Comp. Appl. Math. 41, 33 (2022). https://doi.org/10.1007/s40314-021-01740-y

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  • DOI: https://doi.org/10.1007/s40314-021-01740-y

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