Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 307298, 24 pages
http://dx.doi.org/10.1155/2009/307298
Research Article

Modified Jacobian Newton Iterative Method: Theory and Applications

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

Received 3 September 2008; Revised 10 November 2008; Accepted 4 December 2008

Academic Editor: José Roberto Castilho Piqueira

Copyright © 2009 Jürgen Geiser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. I. Marchuk, “Some applicatons of splitting-up methods to the solution of problems in mathematical physics,” Aplikace Matematiky, vol. 1, pp. 103–132, 1968. View at Google Scholar
  2. G. Strang, “On the construction and comparison of difference schemes,” SIAM Journal on Numerical Analysis, vol. 5, no. 3, pp. 506–517, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Yoshida, “Construction of higher order symplectic integrators,” Physics Letters A, vol. 150, no. 5–7, pp. 262–268, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. F. Kanney, C. T. Miller, and C. T. Kelley, “Convergence of iterative split-operator approaches for approximating nonlinear reactive problems,” Advances in Water Resources, vol. 26, no. 3, pp. 247–261, 2003. View at Publisher · View at Google Scholar
  5. P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithm, vol. 35 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2004. View at Zentralblatt MATH · View at MathSciNet
  6. P. Deuflhard, “A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting,” Numerische Mathematik, vol. 22, no. 4, pp. 289–315, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Ruzicka, Nichtlineare Funktionalanalysis, Springer, Berlin, Germany, 2004. View at Zentralblatt MATH
  8. E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators, Springer, New York, NY, USA, 1990. View at Zentralblatt MATH · View at MathSciNet
  9. J. Geiser, O. Klein, and P. Philip, “Anisotropic thermal conductivity in apparatus insulation: numerical study of effects on the temperature field during sublimation growth of silicon carbide single crystals,” preprint no. 1034 of Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, 2005.
  10. J. Geiser, “Discretization methods with embedded analytical solutions for convection dominated transport in porous media,” in Proceedings of the 3rd International Conference on Numerical Analysis and Its Applications (NAA '04), vol. 3401 of Lecture Notes in Computer Science, pp. 288–295, Springer, Rousse, Bulgaria, June-July 2004.
  11. W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, vol. 33 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 2003. View at Zentralblatt MATH · View at MathSciNet
  12. K. H. Karlsen, K.-A. Lie, J. R. Natvig, H. F. Nordhaug, and H. K. Dahle, “Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies,” Journal of Computational Physics, vol. 173, no. 2, pp. 636–663, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. H. Karlsen and N. H. Risebro, “Corrected operator splitting for nonlinear parabolic equations,” SIAM Journal on Numerical Analysis, vol. 37, no. 3, pp. 980–1003, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. I. Faragó and J. Geiser, “Iterative operator-splitting methods for linear problems,” International Journal of Computational Science and Engineering, vol. 3, no. 4, pp. 255–263, 2007. View at Publisher · View at Google Scholar
  15. C. T. Kelly, Iterative Methods for Linear and Nonlinear Equations, Frontiers in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1995. View at Zentralblatt MATH · View at MathSciNet
  16. J. Geiser, “Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 227–242, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. K. H. Karlsen and N. H. Risebro, “An operator splitting method for nonlinear convection-diffusion equations,” Numerische Mathematik, vol. 77, no. 3, pp. 365–382, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet