Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Finite difference solution of a nonlinear Klein-Gordon equation with an external source


Authors: G. Berikelashvili, O. Jokhadze, S. Kharibegashvili and B. Midodashvili
Journal: Math. Comp. 80 (2011), 847-862
MSC (2010): Primary 65M06, 35L70
DOI: https://doi.org/10.1090/S0025-5718-2010-02416-0
Published electronically: August 25, 2010
MathSciNet review: 2772098
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider the Darboux problem for a (1+1)-dimensional cubic nonlinear Klein-Gordon equation with an external source. Stable finite difference scheme is constructed on a four-point stencil, which does not require additional iterations for passing from one level to another. It is proved, that the finite difference scheme converges with the rate $ O(h^2)$, when the exact solution belongs to the Sobolev space $ W_2^2$.


References [Enhancements On Off] (What's this?)

  • 1. F.J. Alexander and S. Habib, Statistical mechanics of kinds in 1+1 dimensions, Phys. Rev. Lett. 71 (1993), 955-958.
  • 2. G.K. Berikelashvili, O.M. Jokhadze, B.G. Midodashvili and S.S. Kharibegashvili, On existence and nonexistence of global solutions of the first Darboux problem for nonlinear wave equations, Differ. Uravn. 44 (2008), no. 3, 359-372. MR 2437059 (2009c:35299)
  • 3. A.R. Bishop, J.A. Krumhansl and S.E. Trullinger, Solitons in condensed matter: a paradigm. Phys. D 1, no. 1 (1980), 1-44. MR 573367 (81g:82047)
  • 4. A.V. Bitsadze, Some Classes of Partial Differential Equations, Advanced Studies in Contemporary Mathematics, 4 (Gordon and Breach, 1988). MR 1052404 (91h:35001)
  • 5. T. Cazenave, A. Haraux, L. Vazquez and F.B. Weissler, Nonlinear effects in the wave equation with a cubic restoring force, Comput. Mech. 5 (1989) 49-72.
  • 6. G. Darboux, Leçons sur la Théorie Générale des Surfaces et les Applications Géométriques du Calcul Infinitésimal, Vol. III(French) (Gauthier-Villars, 1894).
  • 7. M. Flato, J.C.H. Simon and E. Taflin, Asymptotic completeness, global existence and the infrared problem for the Maxwell-Dirac equations, Mem. Amer. Math. Soc. 127, no. 606, (1997). MR 1407900 (97j:35131)
  • 8. S. Gellerstedt, Sur un Problème aux Limites pour une Équation Linéaire aux Dérivées Partielles du Second Ordre de Type Mixte, (French) (Uppsala University, 1935).
  • 9. O.G. Goman, The equation of a reflected wave, Vestn. Mosk. Univ., Ser. I (Russian) 23 (1968), 84-87.
  • 10. N. Gonzalez, An example of pure stability for the wave equation with moving boundary, J. Math. Anal. Appl. 228 (1998), 51-59. MR 1659940 (99f:35115)
  • 11. É. Goursat, Cours D'analyse Mathématique, Tome III. (French) (Èditions Jacques Gabay, Sceaux, 1992). MR 1296666
  • 12. J. Hadamard, Résolution d'un problème aux limites pour les équations linéaires du type hyperbolique, Bull. Soc. Math. France (French) 32 (1904), 242-268. MR 1504486
  • 13. A. Inoue, Sur $ \square u+u^3=f$ dans un domaine noncylindrique, J. Math. Anal. Appl. (French) 46 (1974), 777-819. MR 0346329 (49:11054)
  • 14. K. Jörgens, Über die nichtlinearen wellengleichungen der mathematischen physik, Math. Annalen 138 (1959) 179-202. MR 0144073 (26:1621)
  • 15. N.A. Lar'kin and M.H. Simões, Nonlinear wave equation with a nonlinear boundary damping in a noncylindrical domain, Mat. Contemp. 23 (2002), 19-34. MR 1965774 (2004c:35286)
  • 16. H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys. 73 (2005), 249-258. MR 2188297 (2006i:35249)
  • 17. B.N. Lu and S.M. Fang, Convergence on finite difference solution of semilinear wave equation in one space variable, Chinese Q. J. Math. 12 (1997), 35-40.
  • 18. J.K. Perring and T.H.R. Skyrme, A model unified field equations, Nuclear Phys. 31 (1962), 550-555. MR 0138393 (25:1840)
  • 19. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Part II: Fourier Analysis, Self-Adjointness (Academic Press, 1975) MR 0493420 (58:12429b)
  • 20. J. Sather, The existence of a global classical solution of the initial-boundary value problem for $ \square u +u^3 =f$, Arch. Rational Mech. Anal. 22 (1966), 292-307. MR 0197965 (33:6124)
  • 21. F. Tricomi, Lectures on Partial Differential Equations (Izdat. Inost. Lit., 1957).
  • 22. S.D. Troitskaya, A boundary-value problem for hyperbolic equations, Izv. Math. (Russian) 62 (1998), 399-428. MR 1623842 (99f:35121)
  • 23. C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM Theory, Comm. Math. Phys. 127 (1990), 479-528. MR 1040892 (91b:58236)
  • 24. Y.L. Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, (International Academic Publishers, Beijing, 1991). MR 1133399 (92m:65001)
  • 25. A.A. Samarski, R.D. Lazarov and V.L. Makarov. Difference schemes for differential equations with generalized solutions. (Russian) Visshaya Shkola, Moscow, 1987.
  • 26. D.B. Duncan, Symplectic finite difference approximations of the nonlinear Klein-Gordon equation, SIAM J. Numer. Anal. 34, no. 5 (1997), 1742-1760. MR 1472194 (98m:65139)
  • 27. K. Feng, D. Wang, A note on conservation laws of symplectic difference schemes for Hamiltonian systems, J. Comput. Math. 9, no.3, (1991), 229-237. MR 1150184 (93b:65113)
  • 28. W. Sha, Z. Huang, X. Wu, M. Chen, Application of the symplectic finite-difference time-domain scheme to electromagnetic simulation, J. Comput. Phys. 225, no. 1, (2007), 33-50. MR 2346670 (2008f:78016)
  • 29. Y. Wang, B. Wang, High-order multi-symplectic schemes for the nonlinear Klein-Gordon equation, Appl. Math. Comput. 166, no. 3, (2005), 608-632. MR 2150493 (2006d:65147)
  • 30. Z. Wang, B. Guo, Legendre rational spectral method for nonlinear Klein-Gordon equation, Numer. Math. J. Chin. Univ. 15, no. 2, (2006), 143-149 . MR 2254925 (2008a:65238)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M06, 35L70

Retrieve articles in all journals with MSC (2010): 65M06, 35L70


Additional Information

G. Berikelashvili
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
Email: bergi@rmi.acnet.ge

O. Jokhadze
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
Email: ojokhadze@yahoo.com

S. Kharibegashvili
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia
Email: khar@rmi.acnet.ge

B. Midodashvili
Affiliation: I. Javakhishvili Tbilisi State University, 2, University str., 0186, Tbilisi, Georgia
Email: bidmid@hotmail.com

DOI: https://doi.org/10.1090/S0025-5718-2010-02416-0
Keywords: Nonlinear Klein-Gordon equation, Darboux problem, finite difference solution
Received by editor(s): May 13, 2008
Received by editor(s) in revised form: February 4, 2010
Published electronically: August 25, 2010
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society