We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments.
Mots-clés : existence, uniqueness, finite difference methods, error estimates
@article{M2AN_2012__46_4_841_0,
author = {Bournaveas, Nikolaos and Zouraris, Georgios E.},
title = {Theory and numerical approximations for a nonlinear 1 + 1 {Dirac} system},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {841--874},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
doi = {10.1051/m2an/2011071},
mrnumber = {2891472},
zbl = {1274.65232},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2011071/}
}
TY - JOUR AU - Bournaveas, Nikolaos AU - Zouraris, Georgios E. TI - Theory and numerical approximations for a nonlinear 1 + 1 Dirac system JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 841 EP - 874 VL - 46 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011071/ DO - 10.1051/m2an/2011071 LA - en ID - M2AN_2012__46_4_841_0 ER -
%0 Journal Article %A Bournaveas, Nikolaos %A Zouraris, Georgios E. %T Theory and numerical approximations for a nonlinear 1 + 1 Dirac system %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 841-874 %V 46 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011071/ %R 10.1051/m2an/2011071 %G en %F M2AN_2012__46_4_841_0
Bournaveas, Nikolaos; Zouraris, Georgios E. Theory and numerical approximations for a nonlinear 1 + 1 Dirac system. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 841-874. doi : 10.1051/m2an/2011071. http://archive.numdam.org/articles/10.1051/m2an/2011071/
[1] , Linearized Crank-Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys. 99 (1992) 348-350. | MR | Zbl
[2] and , Interaction dynamics for the solitary waves of a nonlinear Dirac model. Phys. Lett. A 86 (1981) 327-332. | MR
[3] , and , The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput. 13 (1983) 1-15. | MR | Zbl
[4] , Local and global solutions for a nonlinear Dirac system. Advances Differential Equations 9 (2004) 677-698. | MR | Zbl
[5] , Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension. Discrete Contin. Dyn. Syst. Ser. A 20 (2008) 605-616. | MR | Zbl
[6] , and , Virial identity and weak dispersion for the magnetic Dirac equation. J. Math. Pures Appl. 95 (2011) 137-150. | MR | Zbl
[7] , Estabilidad y convergencia de esquemas numericos para sistemas de Dirac no lineales. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingenieria 5 (1989) 185-202.
[8] and , Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys. 83 (1989) 407-423. | MR | Zbl
[9] , Global solutions of the Cauchy problem for the classical Coupled Maxwell-Dirac and other nonlinear Dirac equations in one space dimension. Proc. Amer. Math. Soc. 69 (1978) 289-296. | MR | Zbl
[10] , Galerkin methods for first order hyperbolics : an example. SIAM J Numer. Anal. 10 (1973) 890-899. | MR | Zbl
[11] , On one-dimensional coupled Dirac equations. Trans. Amer. Math. Soc. 231 (1977) 531-539. | MR | Zbl
[12] , and , Spectral and pseudospectral approximations using Hermite functions : application to the Dirac equation. Adv. Comput. Math. 19 (2003) 35-55. | MR | Zbl
[13] and , Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys. 211 (2006) 448-472. | MR | Zbl
[14] , Lectures on Nonlinear Hyperbolic Differential Equations. Springer-Verlag (1997). | MR | Zbl
[15] , Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput. 64 (1994) 13-45. | MR | Zbl
[16] , Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19 (1967) 508-520. | MR | Zbl
[17] , One dimensional Dirac equation with quadratic nonlinearities. Discrete Contin. Dyn. Syst. Ser. A 13 (2005) 277-290. | MR | Zbl
[18] , Dirac equation with certain quadratic nonlinearities in one space dimension. Commun. Contemp. Math. 9 (2007) 421-435. | MR | Zbl
[19] , and , Small global solutions for nonlinear Dirac equations. Differential Integral Equations 17 (2004) 623-636. | MR | Zbl
[20] , , and , Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219 (2005) 1-20. | MR | Zbl
[21] , and , Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50 (2010) 403-451. | MR | Zbl
[22] , Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci. 53 (1984). | MR | Zbl
[23] and , On a semi-group approach to quantum field equations. Nuovo Cimento A 2 (1971) 122-138. | MR
[24] , Non-linear semi-groups. Ann. of Math. 78 (1963) 339-364. | MR | Zbl
[25] and , Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete Contin. Dyn. Syst. Ser. B 6 (2006) 623-640. | MR | Zbl
[26] , The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, Heidelberg (2010). | MR | Zbl
[27] and , An efficient adaptive mesh redistribution method for a non-linear Dirac equation. J. Comput. Phys. 222 (2007) 176-193. | MR | Zbl
[28] , On the convergence of a linear conservative two-step finite element method for the nonlinear Schrödinger equation. ESAIM : M2AN 35 (2001) 389-405. | Numdam | MR | Zbl
Cité par Sources :
