A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime

Cai, Yongyong; Wang, Yan

doi:10.1051/m2an/2018015

Cai, Yongyong ¹ ; Wang, Yan ¹

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 543-566.

Résumé

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter $ε \in (0, 1]$ inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength $O (ε^{2})$ and $O (1)$ in time and space, respectively. In the nonrelativistic regime, i.e., $0 < ε ⩽ 1$ the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds in $ε \in (0, 1]$ . The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as $h^{m_{0}} + \frac{τ^{2}}{ε^{2}}$ and $h^{m_{0}} + τ^{2} + ε^{2}$ , where $h$ is the mesh size, $τ$ is the time step and $m_{0}$ depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O (τ)$ for all $ε \in (0, 1]$ and optimally with quadratic convergence rate at $O (τ^{2})$ in the regimes when either in the regimes when either $ε = 0 (1)$ or $0 < ε ≲ τ$ . Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

MR Zbl

DOI : 10.1051/m2an/2018015

Classification : 35Q40, 65M70, 65N35, 81W05
Mots-clés : Nonlinear Dirac equation, nonrelativistic limit, uniformly accurate, multiscale time integrator, exponential wave integrator, spectral method, error bound

Affiliations des auteurs :

Cai, Yongyong ¹ ; Wang, Yan ¹

@article{M2AN_2018__52_2_543_0,
     author = {Cai, Yongyong and Wang, Yan},
     title = {A uniformly accurate {(UA)} multiscale time integrator pseudospectral method for the nonlinear {Dirac} equation in the nonrelativistic limit regime},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {543--566},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {2},
     year = {2018},
     doi = {10.1051/m2an/2018015},
     zbl = {1404.35377},
     mrnumber = {3834435},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2018015/}
}

TY  - JOUR
AU  - Cai, Yongyong
AU  - Wang, Yan
TI  - A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 543
EP  - 566
VL  - 52
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2018015/
DO  - 10.1051/m2an/2018015
LA  - en
ID  - M2AN_2018__52_2_543_0
ER  -

%0 Journal Article
%A Cai, Yongyong
%A Wang, Yan
%T A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 543-566
%V 52
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2018015/
%R 10.1051/m2an/2018015
%G en
%F M2AN_2018__52_2_543_0

Cai, Yongyong; Wang, Yan. A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 543-566. doi : 10.1051/m2an/2018015. http://archive.numdam.org/articles/10.1051/m2an/2018015/

Bibliographie
Cité par

[1] A. Alvarez, Linearized Crank–Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys. 99 (1992) 348–350. | DOI | MR | Zbl

[2] A. Alvarez, P. Y. Kuo and L. Vazquez, The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput. 13 (1983) 1–15. | DOI | MR | Zbl

[3] X. Antoine, E. Lorin, J. Sater, F. Fillion-Gourdeau and A.D. Bandrauk, Absorbing boundary conditions for relativistic quantum mechanics equations. J. Comput. Phys. 277 (2014) 268–304. | DOI | MR | Zbl

[4] M. Balabane, T. Cazenave and L. Vazquez, Existence of standing waves for Dirac fields with singular nonlinearities. Commun. Math. Phys. 133 (1990) 53–74. | DOI | MR | Zbl

[5] 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) 1103–1127. | DOI | MR | Zbl

[6] W. Bao and X. Li, An efficient and stable numerical method for the Maxwell–Dirac system. J. Comput. Phys. 199 (2004) 663–687. | DOI | MR | Zbl

[7] W. Bao, Y. Cai and X. Zhao, A uniformly accurate multiscale time integrator pseudospectral method for the Klein–Gordon equation in the non-relativistic limit regime. SIAM J. Numer. Anal. 52 (2014) 2488–2511. | DOI | MR | Zbl

[8] W. Bao, Y. Cai, X. Jia and Q. Tang, A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the non-relativistic limit regime. SIAM J. Numer. Anal. 54 (2016) 1785–1812. | DOI | MR | Zbl

[9] W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the non-relativistic limit regime. Sci. China Math. 59 (2016) 1461–1494. | DOI | MR | Zbl

[10] W. Bao, Y. Cai, X. Jia and Q. Tang, Numerical methods and comparison for the Dirac equation in the non-relativistic limit regime. J. Sci. Comput. 71 (2017) 1094–1134. | DOI | MR | Zbl

[11] T. Bartsch and Y. Ding, Solutions of nonlinear Dirac equations. J. Diff. Equ. 226 (2006) 210–249. | DOI | MR | Zbl

[12] P. Bechouche, N. Mauser and F. Poupaud, (Semi)-non-relativistic limits of the Dirac equation with external time-dependent electromagnetic field. Commun. Math. Phys. 197 (1998) 405–425. | DOI | MR | Zbl

[13] N. Bournaveas and G. E. Zouraris, Theory and numerical approximations for a nonlinear 1 + 1 Dirac system. ESAIM: M2AN 46 (2012) 841–874. | DOI | Numdam | MR | Zbl

[14] D. Brinkman, C. Heitzinger and P. A. Markowich, A convergent 2D finite-difference scheme for the Dirac–Poisson system and the simulation of graphene. J. Comput. Phys. 257 (2014) 318–332. | DOI | MR | Zbl

[15] Y. Cai and Y. Wang, (Semi-)Nonrelativisitic limit of the nonlinear Dirac equations. In preparation.

[16] S. J. Chang, S. D. Ellis and B. W. Lee, Chiral confinement: an exact solution of the massive Thirring model. Phys. Rev. D 11 (1975) 3572–3582. | DOI

[17] P. Chartier, N. Crouseilles, M. Lemou and F. Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein–Gordon and nonlinear Schrödinger equations. Numer. Math. 129 (2015) 211–250. | DOI | MR | Zbl

[18] R. J. Cirincione and P. R. Chernoff, Dirac and Klein Gordon equations: convergence of solutions in the non-relativistic limit. Commun. Math. Phys. 79 (1981) 33–46. | DOI | MR | Zbl

[19] M. Esteban and E. Séré, An overview on linear and nonlinear Dirac equations. Discrete Contin. Dyn. Syst. 8 (2002) 381–397. | DOI | MR | Zbl

[20] E. Faou and K. Schratz, Asymptotic preserving schemes for the Klein–Gordon equation in the non-relativistic limit regime. Numer. Math. 126 (2014) 441–469. | DOI | MR | Zbl

[21] J. Frutos and J. M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys. 83 (1989) 407–423. | DOI | MR | Zbl

[22] W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials. Numer. Math. 3 (1961) 381–397. | DOI | MR | Zbl

[23] L. H. Haddad and L. D. Carr, The nonlinear Dirac equation in Bose–Einstein condensates: foundation and symmetries. Physics D 238 (2009) 1413–21. | DOI | MR | Zbl

[24] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Springer-Verlag (2002). | DOI | MR | Zbl

[25] R. Hammer, W. Pötz and A. Arnold, Single-cone real-space finite difference scheme for the time-dependent Dirac equation. J. Comput. Phys. 265 (2014) 50–70. | DOI | MR | Zbl

[26] M. Hochbruck and A. Ostermann, Exponential integrators. Acta Numer. 19 (2000) 209–286. | DOI | MR | Zbl

[27] 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) 761–789. | DOI | MR | Zbl

[28] M. Lemou, F. Méhats and X. Zhao, Uniformly accurate numerical schemes for the nonlinear Dirac equation in the non-relativistic limit regime. Commun. Math. Sci. 15 (2017) 1107–1128. | DOI | MR | Zbl

[29] N. Masmoudi and K. Nakanishi, From nonlinear Klein–Gordon equation to a system of coupled nonlinear Schrödinger equations. Math. Ann. 324 (2002) 359–389. | DOI | MR | Zbl

[30] P. Mathieu, Soliton solutions for Dirac equations with homogeneous non-linearity in (1 + 1) dimensions. J. Phys. A: Math. Gen. 18 (1985) L1061–L1066. | DOI | MR

[31] M. Merkl, A. Jacob, F. E. Zimmer, P. Öhberg and L. Santos, Chiral confinement in quasirelativistic Bose–Einstein condensates. Phys. Rev. Lett. 104 (2010) 073603. | DOI

[32] F. Merle, Existence of stationary states for nonlinear Dirac equations. J. Diff. Equ. 74 (1988) 50–68. | DOI | MR | Zbl

[33] B. Najman, The non-relativistic limit of the nonlinear Dirac equation. Ann. Inst. Henri Poincaré 9 (1992) 3–12. | DOI | Numdam | MR | Zbl

[34] B. Saha, Nonlinear spinor fields and its role in cosmology. Int. J. Theor. Phys. 51 (2012) 1812–1837. | DOI | MR | Zbl

[35] A. Y. Schoene, On the non-relativistic limits of the Klein–Gordon and Dirac equations. J. Math. Anal. Appl. 71 (1979) 36–47. | DOI | MR | Zbl

[36] S. Shao and H. Tang, Higher-order accurate Runge–Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete Contin. Dyn. Syst. B 6 (2006) 623–640. | MR | Zbl

[37] J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications. Springer-Verlag, Berlin, Heidelberg (2011). | DOI | Zbl

[38] M. Soler, Classical, stable, nonlinear spinor field with positive rest energy. Phys. Rev. D 1 (1970) 2766–2769. | DOI

[39] L. Vazquez, Localised solutions of a non-linear spinor field. J. Phys. A: Math. Gen. 10 (1977) 1361–1368. | DOI | MR

[40] H. Wang and H. Tang, An efficient adaptive mesh redistribution method for a nonlinear Dirac equation. J. Comput.Phys. 222 (2007) 176–193. | DOI | MR | Zbl

[41] H. Wu, Z. Huang, S. Jin and D. Yin, Gaussian beam methods for the Dirac equation in the semi-classical regime. Commun. Math. Sci. 10 (2012) 1301–1315. | DOI | MR | Zbl

[42] J. Xu, S. Shao and H. Tang, Numerical methods for nonlinear Dirac equation. J. Comput. Phys. 245 (2013) 131–149. | DOI | MR | Zbl

Cité par Sources :