Theory and numerical approximations for a nonlinear 1 + 1 Dirac system

Nikolaos Bournaveas; Georgios E. Zouraris

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 841-874
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Bournaveas, Nikolaos, and Zouraris, Georgios E.. "Theory and numerical approximations for a nonlinear 1 + 1 Dirac system." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 841-874. <http://eudml.org/doc/276381>.

@article{Bournaveas2012,
abstract = {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.},
author = {Bournaveas, Nikolaos, Zouraris, Georgios E.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Existence; uniqueness; finite difference methods; error estimates; implicit-explicit finite difference method; nonlinear Dirac system},
language = {eng},
month = {2},
number = {4},
pages = {841-874},
publisher = {EDP Sciences},
title = {Theory and numerical approximations for a nonlinear 1 + 1 Dirac system},
url = {http://eudml.org/doc/276381},
volume = {46},
year = {2012},
}

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
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 841
EP - 874
AB - 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.
LA - eng
KW - Existence; uniqueness; finite difference methods; error estimates; implicit-explicit finite difference method; nonlinear Dirac system
UR - http://eudml.org/doc/276381
ER -

References

top
  1. A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations. J. Comput. Phys.99 (1992) 348–350.  
  2. A. Alvarez and B. Carreras, Interaction dynamics for the solitary waves of a nonlinear Dirac model. Phys. Lett. A86 (1981) 327–332.  
  3. A. Alvarez, Pen-Yu Kuo and L. Vazquez, The numerical study of a nonlinear one-dimensional Dirac equation. Appl. Math. Comput.13 (1983) 1–15.  
  4. N. Bournaveas, Local and global solutions for a nonlinear Dirac system. Advances Differential Equations9 (2004) 677–698.  
  5. N. Bournaveas, Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension. Discrete Contin. Dyn. Syst. Ser. A20 (2008) 605–616.  
  6. N. Boussaid, P. D’Ancona and L. Fanelli, Virial identity and weak dispersion for the magnetic Dirac equation. J. Math. Pures Appl.95 (2011) 137–150.  
  7. J. De Frutos, Estabilidad y convergencia de esquemas numericos para sistemas de Dirac no lineales. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingenieria5 (1989) 185–202.  
  8. J. De Frutos and J.M. Sanz-Serna, Split-step spectral schemes for nonlinear Dirac systems. J. Comput. Phys.83 (1989) 407–423.  
  9. V. Delgado, 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.  
  10. T. Dupont, Galerkin methods for first order hyperbolics : an example. SIAM J Numer. Anal.10 (1973) 890–899.  
  11. R.T. Glassey, On one-dimensional coupled Dirac equations. Trans. Amer. Math. Soc.231 (1977) 531–539.  
  12. B.-Y. Guo, J. Shen and C.-L. Xu, Spectral and pseudospectral approximations using Hermite functions : application to the Dirac equation. Adv. Comput. Math.19 (2003) 35–55.  
  13. J. Hong and C. Li, Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations. J. Comput. Phys.211 (2006) 448–472.  
  14. L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations. Springer-Verlag (1997).  
  15. S. Jiménez, Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput.64 (1994) 13–45.  
  16. T. Kato, Nonlinear semigroups and evolution equations. J. Math. Soc. Japan19 (1967) 508–520.  
  17. S. Machihara, One dimensional Dirac equation with quadratic nonlinearities. Discrete Contin. Dyn. Syst. Ser. A13 (2005) 277–290.  
  18. S. Machihara, Dirac equation with certain quadratic nonlinearities in one space dimension. Commun. Contemp. Math.9 (2007) 421–435.  
  19. S. Machihara, M. Nakamura and T. Ozawa, Small global solutions for nonlinear Dirac equations. Differential Integral Equations17 (2004) 623–636.  
  20. S. Machihara, M. Nakamura, K. Nakanishi and T. Ozawa, Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal.219 (2005) 1–20.  
  21. S. Machihara, K. Nakanishi and K. Tsugawa, Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math.50 (2010) 403–451.  
  22. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Appl. Math. Sci.53 (1984).  
  23. E. Salusti and A. Tesei, On a semi-group approach to quantum field equations. Nuovo Cimento A2 (1971) 122–138.  
  24. I. E. Segal, Non-linear semi-groups. Ann. of Math.78 (1963) 339–364.  
  25. S. Shao and H. Tang, Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model. Discrete Contin. Dyn. Syst. Ser. B6 (2006) 623–640.  
  26. B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, Heidelberg (2010).  
  27. H. Wang and H. Tang, An efficient adaptive mesh redistribution method for a non-linear Dirac equation. J. Comput. Phys.222 (2007) 176–193.  
  28. G.E. Zouraris, On the convergence of a linear conservative two-step finite element method for the nonlinear Schrödinger equation. ESAIM : M2AN35 (2001) 389–405.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.