An energy-preserving exponential scheme with scalar auxiliary variable approach for the nonlinear Dirac equations

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this work, an energy-preserving scheme is proposed for the nonlinear Dirac equation by combining the exponential time differencing method with the scalar auxiliary variable (SAV) approach. First, the original equations can be transformed into the equivalent systems by utilizing the SAV technique. Then the exponential time integrator method is applied for discretizing the temporal derivative, and the standard central difference scheme is used for approximating the spatial derivative for the equivalent one. Finally, the reformulated systems, the semi-discrete spatial scheme, and the fully-discrete, linearly implicit exponential scheme are proven to be energy conserving. The numerical experiments confirm the theoretical results.

Keywords

nonlinear Dirac equation / exponential time integrator method / scalar auxiliary variable / energy conserving

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] J. Zhang, Y. Zhang, An infinite sequence of localized semiclassical states for nonlinear Maxwell-Dirac system, J. Geom. Anal., 34 (2024), 277. https://doi.org/10.1007/s12220-024-01724-4

[2] N. S. Papageorgiou, J. Zhang, W. Zhang, Global existence and multiplicity of solutions for nonlinear singular eigenvalue problems, Discrete Contin. Dyn. Syst.-S, (2024), 1-17. https://doi.org/10.3934/dcdss.2024018

[3] J. Y. Li, T. C. Wang, Optimal point-wise error estimate of two conservative fourth-order compact finite difference schemes for the nonlinear Dirac equation, Appl. Numer. Math., 162 (2021), 150-170. https://doi.org/10.1016/j.apnum.2020.12.010

[4] Y. Feng, Z. G. Xu, J. Yin, Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials, Appl. Numer. Math., 172 (2022), 50-66. https://doi.org/10.1016/j.apnum.2021.09.018

[5] W. Z. Bao, Y. Y. Cai, X. W. Jia, Q. L. Tang, Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime, J. Sci. Comput., 71 (2017), 1094-1134. https://doi.org/10.1007/s10915-016-0333-3

[6] M. Lemou, F. Méhats, X. F. Zhao, Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime, Commun. Math. Sci., 15 (2017), 1107-1128. https://doi.org/10.4310/CMS.2017.v15.n4.a10

[7] K. Schratz, Y. Wang, X. F. Zhao, Low-regularity integrators for nonlinear Dirac equations, Math.Comput., 90 (2021), 189-214. https://doi.org/10.1090/mcom/3557

[8] Y. Zhe, J. Shen, Efficient SAV-Hermite methods for the nonlinear Dirac equation, Discrete Contin. Dyn. Syst.-Ser. B, 28 (2023), 3428-3452. https://doi.org/10.3934/dcdsb.2022225

[9] R. Yang, Y. Xing, Energy conserving discontinuous Galerkin method with scalar auxiliary variable technique for the nonlinear Dirac equation, J. Comput. Phys., 463 (2022), 111278. https://doi.org/10.1016/j.jcp.2022.111278

[10] F. Fillion-Gourdeau, E. Lorin, A. D. Bandrauk, Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling, Comput. Phys. Commun., 183 (2012), 1403-1415. https://doi.org/10.1016/j.cpc.2012.02.012

[11] X. H. Yang, W. L. Qiu, H. X. Zhang, L. Tang, An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation, Comput. Math. Appl., 102 (2021), 233-247. https://doi.org/10.1016/j.camwa.2021.10.021

[12] Z. Y. Chen, H. X. Zhang, H. Chen, ADI compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann-Liouville integral kernels, Fractal Fractional, 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707

[13] H. X. Zhang, X. H. Yang, Y. L. Liu, Y. Liu, An extrapolated CN-WSGD OSC method for a nonlinear time fractional reaction-diffusion equation, Appl. Numer. Math., 157 (2020), 619-633. https://doi.org/10.1016/j.apnum.2020.07.017

[14] X. H. Yang, Z. M. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7

[15] X. H. Yang, H. X. Zhang, Q. Zhang, G. W. Yuan, Simple positivity preserving nonlinear finite volume scheme for subdiffusion equations on general non-conforming distorted meshes, Nonlinear Dyn., 108 (2022), 3859-3886. https://doi.org/10.1007/s11071-022-07399-2

[16] C. J. Li, H. X. Zhang, X. H. Yang, A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK, Comput. Math. Appl., 176 (2024), 349-370. https://doi.org/10.1016/j.camwa.2024.11.006

[17] Y. Shi, X. H. Yang, Z. M. Zhang, Construction of a new time-space two-grid method and its solution for the generalized Burgers’ equation, Appl. Math. Lett., 158 (2024), 109244. https://doi.org/10.1016/j.aml.2024.109244

[18] X. H. Yang, W. Wang, Z. Y. Zhou, H. X. Zhang, An efficient compact difference method for the fourth-order nonlocal subdiffusion problem, Taiwan. J. Math., 1 (2024), 1-32. https://doi.org/10.11650/tjm/240906

[19] X. H. Yang, W. L. Qiu, H. F. Chen, H. X. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497-513. https://doi.org/10.1016/j.apnum.2021.11.004

[20] X. H. Yang, Z. M. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024), 62. https://doi.org/10.1007/s10915-024-02616-z

[21] H. X. Zhang, X. H. Yang, D. Xu, Unconditional convergence of linearized orthogonal spline collocation algorithm for semilinear subdiffusion equation with nonsmooth solution, Numer. Methods Partial Differ. Equations, 37 (2021), 1361-1373. https://doi.org/10.1002/num.22583

[22] X. H. Yang, H. X. Zhang, The uniform l¹ long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data , Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644

[23] D. F. Li, X. X. Li, Relaxation exponential Rosenbrock-type methods for oscillatory Hamiltonian systems, SIAM J. Sci. Comput., 45 (2023), A2886-A2911. https://doi.org/10.1137/22M1511345

[24] D. F. Li, X. X. Li, J. Yang, Relaxation exponential Runge-Kutta methods and their applications to semilinear dissipative/conservative systems, Commun. Comput. Phys., 36 (2024), 908-942. https://doi.org/10.4208/cicp.OA-2023-0241

[25] D. F. Li, W. W. Sun, Linearly implicit and high-order energy-conserving schemes for nonlinear wave equations, J. Sci. Comput., 83 (2020), 1-17. https://doi.org/10.1007/s10915-020-01245-6

[26] G. Akrivis, B. Y. Li, D. F. Li, Energy-decaying extrapolated RK-SAV methods for the Allen-Cahn and Cahn-Hilliard equations, SIAM J. Sci. Comput., 41 (2019), A3703-A3727. https://doi.org/10.1137/19M1264412

[27] Y. R. Zhang, J. Shen, A generalized SAV approach with relaxation for dissipative systems, J. Comput. Phys., 464 (2022), 111311. https://doi.org/10.1016/j.jcp.2022.111311

[28] J. Xu, S. H. Shao, H. Z. Tang, Numerical methods for nonlinear Dirac equation, J. Comput. Phys., 245 (2013), 131-149. https://doi.org/10.1016/j.jcp.2013.03.031

[29] Y. Y. Fu, D. D. Hu, G. G. Zhang, Arbitrary high-order exponential integrators conservative schemes for the nonlinear Gross-Pitaevskii equation, Comput. Math. Appl., 121 (2022), 102-114. https://doi.org/10.1016/j.camwa.2022.07.004

[30] S. C. Li, X. G. Li, High-order compact methods for the nonlinear Dirac equation, Comput. Appl. Math., 37 (2018), 6483-6498. https://doi.org/10.1007/s40314-018-0705-4

[31] W. Z. Bao, J. Yin, A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation, Res. Math. Sci., 6 (2019), 1-35. https://doi.org/10.1007/s40687-018-0173-x

[32] S. C. Li, X. G. Li, High-order conservative schemes for the nonlinear Dirac equation, Int. J. Comput. Math., 97 (2020), 2355-2374. https://doi.org/10.1080/00207160.2019.1698735

[33] C. L. Jiang, Y. S. Wang, W. J. Cai, A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation, J. Comput. Phys., 419 (2020), 109690. https://doi.org/10.1016/j.jcp.2020.109690