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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 420608, 22 pages
http://dx.doi.org/10.1155/2011/420608
Research Article

A New High-Order Approximation for the Solution of Two-Space-Dimensional Quasilinear Hyperbolic Equations

1Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India
2Department of Mathematics, Aditi Mahavidyalaya, University of Delhi, Delhi 110 039, India

Received 31 March 2011; Revised 13 June 2011; Accepted 5 July 2011

Academic Editor: Ricardo Weder

Copyright © 2011 R. K. Mohanty and Suruchi Singh. 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. M. Ciment and S. H. Leventhal, “Higher order compact implicit schemes for the wave equation,” Mathematics of Computation, vol. 29, no. 132, pp. 985–994, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. Ciment and S. H. Leventhal, “A note on the operator compact implicit method for the wave equation,” Mathematics of Computation, vol. 32, no. 141, pp. 143–147, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. T. Schwartzkopff, C. D. Munz, and E. F. Toro, “Ader: a High-Order Approach for Linear Hyperbolic Systems in 2D,” Journal of Scientific Computing, vol. 17, pp. 231–240, 2002.
  4. T. Schwartzkopff, M. Dumbser, and C. -D. Munz, “Fast high order ADER schemes for linear hyperbolic equations,” Journal of Computational Physics, vol. 197, no. 2, pp. 532–539, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. Dumbser and M. Käser, “Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems,” Journal of Computational Physics, vol. 221, no. 2, pp. 693–723, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. R. K. Mohanty, M. K. Jain, and K. George, “High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients,” Journal of Computational and Applied Mathematics, vol. 70, no. 2, pp. 231–243, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. K. Mohanty, U. Arora, and M. K. Jain, “Linear stability analysis and fourth-order approximations at first time level for the two space dimensional mildly quasi-linear hyperbolic equations,” Numerical Methods for Partial Differential Equations, vol. 17, no. 6, pp. 607–618, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  8. R. K. Mohanty and M. K. Jain, “An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 6, pp. 983–993, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. R. K. Mohanty, “An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 799–806, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Dehghan and A. Shokri, “A numerical method for solving the hyperbolic telegraph equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 4, pp. 1080–1093, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. Dehghan and A. Mohebbi, “High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation,” Numerical Methods for Partial Differential Equations, vol. 25, no. 1, pp. 232–243, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. M. Dehghan and A. Shokri, “A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions,” Numerical Methods for Partial Differential Equations, vol. 25, no. 2, pp. 494–506, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. R. K. Mohanty, “New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations,” International Journal of Computer Mathematics, vol. 86, no. 12, pp. 2061–2071, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Liu and K. Tang, “A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation,” International Journal of Computer Mathematics, vol. 87, no. 10, pp. 2259–2267, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. Y. Hu and H. W. Liu, “An unconditionally stable spline difference scheme for solving the second 2D linear hyperbolic equation,” in Proceedings of the 2nd International Conference on Computer Modeling and Simulation (ICCMS '10), vol. 4, pp. 375–378, 2010. View at Publisher · View at Google Scholar
  16. S. Karaa, “Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 3030–3038, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. R. K. Mohanty and S. Singh, “High accuracy Numerov type discretization for the solution of one-space dimensional non-linear wave equations with variable coefficients,” Journal of Advanced Research in Computer Science, vol. 3, pp. 53–66, 2011.
  18. M. Lees, “Alternating direction methods for hyperbolic differential equations,” vol. 10, pp. 610–616, 1962. View at Zentralblatt MATH
  19. A. R. Gourlay and A. R. Mitchell, “A classification of split difference methods for hyperbolic equations in several space dimensions,” SIAM Journal on Numerical Analysis, vol. 6, pp. 62–71, 1969. View at Publisher · View at Google Scholar
  20. C. T. Kelly, Iterative Methods for Linear and Non-Linear Equations, SIAM Publications, Philadelphia, Pa, USA, 1995.
  21. Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 2nd edition, 2003.
  22. L. A. Hageman and D. M. Young, Applied Iterative Methods, Dover Publications Inc., Mineola, NY, USA, 2004.
  23. R. K. Mohanty, “Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1683–1690, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M. M. Chawla, “Superstable two-step methods for the numerical integration of general second order initial value problems,” Journal of Computational and Applied Mathematics, vol. 12, pp. 217–220, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet