180
Views
9
CrossRef citations to date
0
Altmetric
Research Article

Error analysis of a time fourth-order exponential wave integrator Fourier pseudo-spectral method for the nonlinear Dirac equation

Pages 791-807 | Received 10 Oct 2020, Accepted 03 May 2021, Published online: 08 Jun 2021

References

  • R.A. Adams and J.J. Fournier, Sobolev Spaces, Elsevier, New York, 2003.
  • W. Bao and Y. Cai, Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal. 52 (2014), pp. 1103–1127.
  • W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, Sci. China Math. 59 (2016), pp. 1461–1494.
  • W. Bao and X. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math. 120 (2012), pp. 189–229.
  • W. Bao, X. Dong and X. Zhao, An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, SIAM J. Sci. Comput. 35 (2013), pp. 2903–2927.
  • E. Bas, R. Ozarslan, D. Baleanu and A. Ercan, Comparative simulations for solutions of fractional Sturm-Liouville problems with non-singular operators, Adv. Differ. Equ. 350 (2018), pp. 1–19.
  • X. Dong, A trigonometric integrator pseudospectral discretization for the N-coupled nonlinear Klein-Gordon equations, Numer. Algorithms 62 (2013), pp. 325–336.
  • X. Dong, Stability and convergence of trigonometric integrator pseudospectral discretization for N-coupled nonlinear Klein-Gordon equations, Appl. Math. Comput. 232 (2014), pp. 752–765.
  • J. Dolbeault, M.J. Esteban and E. Séré, On the eigenvalues of operators with gaps: applications to Dirac operator, J. Funct. Anal. 174 (2000), pp. 208–226.
  • J.M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Commun. Math. Phys. 171 (1995), pp. 323–350.
  • J.M. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations, Discrete Contin. Dyn. Syst. 8 (2002), pp. 381–397.
  • R. Hammer, W. Pötz and A. Arnold, A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in (1+1)D, J. Comput. Phys. 256 (2014), pp. 728–747.
  • M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer. 19 (2000), pp. 209–286.
  • J. Hong and C. Li, Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations, J. Comput. Phys. 211 (2006), pp. 448–472.
  • Z. Huang, S. Jin, P.A. Markowich, C. Sparber and C. Zheng, A time-splitting spectral scheme for the Maxwell-Dirac system, J. Comput. Phys. 208 (2005), pp. 761–789.
  • A. Komech and A. Komech, Golbal attraction to solitary waves for a nonlinear Dirac equation with mean field interaction, SIAM J. Math. Anal. 42 (2010), pp. 2944–2964.
  • J. Li and Y. Gao, Energy-preserving trigonometrically-fitted continuous stage Runge-Kutta-Nyström methods for oscillatory hamiltonian systems, Numer. Algorithms 81 (2019), pp. 1379–1401.
  • S. Li, X. Li and F. Shi, Time-splitting methods with charge conservation for the nonlinear Dirac equation, Numer. Meth. Part. D. E. 33 (2017), pp. 1582–1602.
  • J. Li and T. 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), pp. 150–170.
  • J. Li and T. Wang, Analysis of a conservative fourth-order compact finite difference scheme for the KleinCGordonCDirac equation, Comput. Appl. Math. 40 (2021), pp. 114.
  • J. Li and X. Wu, Energy-preserving continuous stage extended Runge-Kutta-Nyström methods for oscillatory hamiltonian systems, Appl. Numer. Math. 145 (2019), pp. 469–487.
  • J.W. Nraun, Q. Su and R. Grobe, Numerical approach to solve the time-dependent Dirac equation, Phys. Rev. A. 59 (1999), pp. 604–612.
  • K.M. Owolabi, High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology, Chaos. Soliton. Fract. 134 (2020), pp. 109723.
  • K.M. Owolabi, B. Karaagac and D. Baleanu, Pattern formation in superdiffusion predator-prey-like problems with integer- and noninteger-order derivatives, 44  (2021), pp. 4018–4036. DOI:https://doi.org/10.1002/mma.7007.
  • K.M. Owolabi and K.C. Patidar, Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput. 240 (2014), pp. 30–50.
  • K.M. Owolabi, E. Pindza and M. Davison, Dynamical study of two predators and one prey system with fractional Fourier transform method, Numer. Meth. Part. D. E. 34 (2018), pp. 1614–1636.
  • E. Pindza and K.M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun. Nonlinear. Sci. 40 (2016), pp. 112–128.
  • J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011.
  • C. Su and W. Yao, A deuflhard-type exponential integrator fourier pseudo-spectral method for the good boussinesq equation, 83 (2020), pp. 4.
  • Z. Xu, X. Dong and Y. Yuan, Error estimates in the energy space for a Gautschi-type integrator spectral discretization for the coupled nonlinear Klein-Gordon equations, J. Comput. Appl. Math. 292 (2016), pp. 402–416.
  • J Xu, S. Shao, H. Tang and D. Wei, Multi-hump solitary waves of a nonlinear Dirac equation, Commun. Math. Sci. 13 (2015), pp. 1219–1242.
  • X.L L.Zhang, Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher–CKolmogorov equation, Appl. Numer. Math. 131 (2018), pp. 39–53.
  • X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, Numer. Methods Partial Differ. Equ. 32 (2016), pp. 266–291.
  • X. Zhao, An exponential wave integrator pseudospectral method for the symmetric regularized-long-wave equation, J. Comput. Math. 34 (2016), pp. 49–69.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.