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Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime

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Abstract

We consider the Klein–Gordon equation in the non-relativistic limit regime, i.e. the speed of light \(c\) tending to infinity. We construct an asymptotic expansion for the solution with respect to the small parameter depending on the inverse of the square of the speed of light. As the first terms of this asymptotic can easily be simulated our approach allows us to construct numerical algorithms that are robust with respect to the large parameter \(c\) producing high oscillations in the exact solution.

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References

  1. 1.

    Bambusi, D.: Birkhoff normal form for some nonlinear PDEs. Commun. Math. Phys. 234, 253–283 (2003)

  2. 2.

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

  3. 3.

    Bourgain, J.: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations. Geom. Funct. Anal. 6, 201–230 (1996)

  4. 4.

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

  5. 5.

    Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. Wiley-Interscience, New York (1959)

  6. 6.

    Brenner, P., Wahl, W.: Global classical solutions of nonlinear wave equations. Math. Z. 176, 87–121 (1981)

  7. 7.

    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2003)

  8. 8.

    Cohen, D., Hairer, E., Lubich, C.: Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations. Numer. Math. 110, 113–143 (2008)

  9. 9.

    Engquist, B., Fokas, A., Hairer, E., Iserles, A.: Highly Oscillatory Problems. Cambridge University Press, Cambridge (2009)

  10. 10.

    Faou, E.: Geometric numerical integration and Schrödinger equations. Zurich lectures in advanced mathematics. European Mathematical Society (EMS), Zürich (2012)

  11. 11.

    Faou, E., Grébert, B.: Hamiltonian interpolation of splitting approximations for nonlinear PDEs. Found. Comput. Math. 11, 381–415 (2011)

  12. 12.

    Faou, E., Grébert, B., Paturel, E.: Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part I: finite dimensional discretization. Numer. Math. 114, 429–458 (2010)

  13. 13.

    Faou, E., Grébert, B., Paturel, E.: Birkhoff normal form for splitting methods applied to semi linear Hamiltonian PDEs. Part II: abstract splitting. Numer. Math. 114, 459–490 (2010)

  14. 14.

    Ginibre, J., Velo, G.: The global Cauchy problem for the non linear Klein–Gordon equation. Math. Z. 189, 487–505 (1985)

  15. 15.

    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2001)

  16. 16.

    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)

  17. 17.

    Hochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)

  18. 18.

    Jiménez, S., Vazquez, L.: Analysis of four numerical schemes for a nonlinear Klein–Gordon equation. Appl. Math. Comput. 35, 61–94 (1990)

  19. 19.

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

  20. 20.

    Machihara, S., Masmoudi, N., Nakanishi, K.: Nonrelativistic limit in the energy space for nonlinear Klein–Gordon equations. Math. Ann. 322, 603–621 (2002)

  21. 21.

    Masmoudi, N., Nakanishi, K.: From nonlinear Klein–Gordon equation to a system of coupled nonlinear Schrödinger equations. Math. Ann. 324, 359–389 (2002)

  22. 22.

    Pascual, P.J., Jiménez, S., Vazquez, L.: Numerical simulations of a nonlinear Klein–Gordon model. Lect. Notes Phys. 448, 211–270 (1995)

  23. 23.

    Petzold, L.R., Jay, L.O., Yen, J.: Numerical solution of highly oscillatory ordinary differential equations. Acta Numerica, 6, 437–483 (1997)

  24. 24.

    Sakurai, J.J.: Advanced Quantum Mechanics. Addison Wesley, Reading (1967)

  25. 25.

    Strauss, W., Vazquez, L.: Numerical solution of a nonlinear Klein–Gordon equation. J. Comput. Phys. 28, 271–278 (1978)

  26. 26.

    Tsutsumi, M.: Nonrelativistic approximation of nonlinear Klein–Gordon equations in two space dimensions. Nonlinear Anal. 8, 637–643 (1984)

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Acknowledgments

We are grateful to Christian Lubich for his helpful comments, and to Markus Penz for fruitful discussions during the preparation of this work.

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Correspondence to Erwan Faou.

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Faou, E., Schratz, K. Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime. Numer. Math. 126, 441–469 (2014). https://doi.org/10.1007/s00211-013-0567-z

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Mathematics Subject Classification (2000)

  • 35C20
  • 65M12
  • 35L05