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Optimal Resolution Methods for the Klein–Gordon–Dirac System in the Nonrelativistic Limit Regime

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Abstract

We propose and compare numerically spatial/temporal resolution of various efficient numerical methods for solving the Klein–Gordon–Dirac system (KGD) in the nonrelativistic limit regime. The KGD system involves a small dimensionless parameter \(0<\varepsilon \ll 1\) in this limit regime and admits rapid oscillations in time as \(\varepsilon \rightarrow 0^+\). By adopting the Fourier spectral discretization for spatial derivatives followed with the time-splitting or exponential wave integrators based on some efficient quadrature rules in phase field, we propose four different numerical discretizations for the KGD system. The discretizations are all fully explicit and valid in one, two and three dimensions. Extensive numerical results demonstrate that these discretizations provide optimal numerical resolutions for the KGD system, i.e., under the mesh strategies \(\tau =O(\varepsilon ^2)\) and \(h=O(1)\) with time step \(\tau \) and mesh size h in terms of \(\varepsilon \), they all perform well with uniform spectral accuracy in space and second-order accuracy in time. In addition, the \(\varepsilon \)-scalability of the best method is improved as \(\tau =O(\varepsilon )\), which is much superior than that of the finite difference methods. For applications, we profile the dynamics of the KGD system in 2D with a honeycomb lattice potential, which depend greatly on the singular perturbation \(\varepsilon \) and the weak/strong interaction.

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References

  1. Bachelot, A.: Global Existence of Large Amplitude Solutions for Dirac–Klein–Gordon Systems in Minkowski Space. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  2. Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose–Einstein condensation. Kinet. Relat. Models 6(1), 1–135 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, W., Cai, Y.: 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(3), 1103–1127 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, W., Cai, Y., Jia, X., Tang, Q.: A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 52(5), 1–41 (2015)

    Google Scholar 

  5. Bao, W., Cai, Y., Jia, X., Tang, Q.: Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime. J. Sci. Comput. 71, 1094–1134 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, W., Cai, Y., Jia, X., Yin, J.: Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime. Sci. China Math. 59(8), 1–34 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bao, W., Cai, Y., Zhao, X.: A uniformly accurate multiscale time integrator pseudospectral method for the Klein–Gordon equation in the nonrelativistic limit regime. SIAM J. Numer. Anal. 52(5), 2488–2511 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bao, W., Dong, X.: Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime. Numer. Math. 120(2), 189–229 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bao, W., Dong, X., Zhao, X.: An exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system. SIAM J. Sci. Comput. 35(6), A2903–A2927 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bao, W., Li, X.: An efficient and stable numerical method for the Maxwell–Dirac system. J. Comput. Phys. 199, 663–687 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bao, W., Shen, J.: A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26, 2010–2028 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bao, W., Zhao, X.: A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein-Gordon–Schrödinger equations in the nonrelativistic limit regime. Numer. Math. 52(5), 1–41 (2015)

    Google Scholar 

  13. Baumstark, S., Faou, E., Schratz, K.: Uniformly accurate exponential-type integrators for Klein–Gordon equations with asymptotic convergence to the classical NLS splitting. Math. Comput. 87, 1227–1254 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bjorken, J.D., Drell, S.D.: Relativistic Quantum Fields. Mcgraw-Hill, Inc, New York (1965)

    MATH  Google Scholar 

  15. Bournaveas, N.: Local existence of energy class solutions for the Dirac–Klein–Gordon equations. Commun. Partial Differ. Equ. 24, 1167–1193 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bournaveas, N.: Low regularity solutions of the Dirac–Klein–Gordon equations in two space dimensions. Commun. Partial Differ. Equ. 26, 1345–1366 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cai, Y., Yi, W.: Error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system in the nonrelativistic limit regime. Commun. Math. Sci. 16, 1325–1346 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chadam, M.J., Glassey, R.T.: On certain global solutions of the Cauchy problem for the (classical) coupled Klein–Gordon–Dirac equations in one and three space dimensions. Arch. Ration. Mech. Anal. 54(3), 223–237 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, G., Zheng, Y.: Solitary waves for the Klein–Gordon–Dirac model. J. Differ. Equ. 7(7), 2263–2284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ding, Y., Xu, T.: On the concentration of semi-classical states for a nonlinear Dirac–Klein–Gordon system. J. Differ. Equ. 256, 1264–1294 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dirac, P.A.M.: Principles of Quantum Mechanics. Oxford University Press, London (1958)

    MATH  Google Scholar 

  22. Dong, X., Xu, Z., Zhao, X.: On time-splitting pseudospectral discretization for nonlinear Klein–Gordon equation in nonrelativistic limit regime. Commun. Comput. Phys. 16(2), 440–466 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Esteban, M.J., Georgiev, V., Séré, E.: Bound-state solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac systems. Lett. Math. Phys. 38(2), 217–220 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fang, Y.F.: A direct proof of global existence for the Dirac–Klein–Gordon equations in one space dimension. Taiwan. J. Math. 8(1), 33–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gautschi, W.: Numerical integration of ordinary differential equations based on trigonometric. Numer. Math. 3, 381–397 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  26. Greiner, W.: Relativistic Quantum Mechanics-Wave Equations. Springer, Berlin (1994)

    Google Scholar 

  27. Grimm, V., Hochbruck, M.: Error analysis of exponential integrators for oscillatory second-order differential equations. J. Phys. A 39, 5495–5507 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 402–426 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Holten, J.W.V.: On the electrodynamics of spinning particles. Nucl. Phys. B 356(1), 3–26 (1991)

    Article  Google Scholar 

  31. Huang, Z., Jin, S., Markowich, P.A., Sparber, C., Zheng, C.: A time-splitting spectral scheme for the Maxwell–Dirac system. J. Comput. Phys. 208(2), 761–789 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lemou, M., Méhats, F., Zhao, X.: Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime. Commun. Math. Sci. 15(4), 1107–1128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lubich, C.: On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77, 2141–2153 (2008)

    Article  MATH  Google Scholar 

  34. Machihara, S., Omoso, T.: The explicit solutions to the nonlinear Dirac equation and Dirac–Klein–Gordon equation. Ric. Mat. 56(1), 19–30 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ohlsson, T.: Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  36. Selberg, S., Tesfahun, A.: Low regularity well-posedness of the Dirac–Klein–Gordon equations in one space dimension. Commun. Contemp. Math. 10(2), 347–353 (2006)

    MathSciNet  MATH  Google Scholar 

  37. Slawianowski, J.J., Kovalchuk, V.: Klein–Gordon–Dirac equation: physical justification and quantization attempts. Rep. Math. Phys. 49, 249–257 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Strang, G.: On the construction and comparision of difference schemes. SIAM J. Numer. Anal. 5, 505–517 (1968)

    Article  Google Scholar 

  39. Su, C.: Comparison of numerical methods for the Zakharov system in the subsonic limit regime. J. Comput. Appl. Math. 330, 441–455 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yi, W., Cai, Y.: Optimal error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system. IMA J. Numer. Anal. (2018). https://doi.org/10.1093/imanum/dry084

    Article  MATH  Google Scholar 

  41. Zhao, X.: On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system. Numer. Methods Partial Differ. Equ. 32(1), 266–291 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Professor Weizhu Bao and Professor Yongyong Cai for their valuable suggestions and comments. Part of this work was done when the authors were visiting the Department of Mathematics at the National University of Singapore in 2017.

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Authors and Affiliations

  1. Institute of Mathematics, College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    Wenfan Yi

  2. Beijing Computational Science Research Center, No. 10 West Dongbeiwang Road, Haidian District, Beijing, 100193, China

    Wenfan Yi

  3. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Xinran Ruan

  4. Laboratoire J.-L. Lions, Université Pierre et Marie Curie, 75252, Paris cedex 05, France

    Xinran Ruan

  5. Department of Mathematics, University of Innsbruck, 6020, Innsbruck, Austria

    Chunmei Su

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  1. Wenfan Yi
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Correspondence to Xinran Ruan.

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This work was supported by the Fundamental Research Funds for the Central Universities 531107051208 and the NSFC Grant 11601148.

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Yi, W., Ruan, X. & Su, C. Optimal Resolution Methods for the Klein–Gordon–Dirac System in the Nonrelativistic Limit Regime. J Sci Comput 79, 1907–1935 (2019). https://doi.org/10.1007/s10915-019-00919-0

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