The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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AN EFFICIENT SECOND-ORDER NON-ITERATIVE FINITE DIFFERENCE SCHEME FOR HYPERBOLIC TELEGRAPH EQUATIONS

  • Jun, Young-Bae (Department of Applied Mathematics, Kumoh National Institute of Technology) ;
  • Hwang, Hong-Taek (Department of Applied Mathematics, Kumoh National Institute of Technology)
  • Received : 2010.04.08
  • Accepted : 2010.11.20
  • Published : 2010.11.30
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Abstract

In this paper, we propose a second-order prediction/correction (SPC) domain decomposition method for solving one dimensional linear hyperbolic partial differential equation $u_{tt}+a(x,t)u_t+b(x,t)u=c(x,t)u_{xx}+{\int}(x,t)$. The method can be applied to variable coefficients problems and singular problems. Unconditional stability and error analysis of the method have been carried out. Numerical results support stability and efficiency of the method.

Keywords

References

  1. W.F. Ames: Numerical methods for partial differential equations. Academic Press, 1992.
  2. J.J. Benito, F. Urena & L. Gravete: Solving parabolic and hyperbolic equations by the generalized finite difference method. J. Comput. Appl. Math. 209 (2007), 208-233. https://doi.org/10.1016/j.cam.2006.10.090
  3. R.L. Burden & J.D. Faires: Numerical Analysis, Thomson Brooks/Cole, 2005.
  4. R. Codina: Finite element approximation of the hyperbolic wave equation in mixed form. Comput. Methods Appl. Mech. Engrg. 197 (2008), 1305-1322. https://doi.org/10.1016/j.cma.2007.11.006
  5. C.N. Dawson & T.F. Dupont: Noniterative domain decomposition for second order hyperbolic problems. Contemp. Math. 157 (1994), 45-52.
  6. H. Ding & Y. Zhang: A new unconditionally stable compact difference scheme of O$({\tau}^2+h^4)$ for the ID linear hyperbolic equation. Appl. Math. Comput. 207 (2009), 236-241. https://doi.org/10.1016/j.amc.2008.10.024
  7. M.J. Gander & L. Halpern: Absorbing boundary conditions for the wave equation and parallel computing. Math. Comp. 74 (2005), 153-176.
  8. X. He & T. Lu: A finite element splitting extrapolation for second order hyperbolic equations. SIAM J. Sci. Comput. 31 (2009/10), 4244-4265.
  9. Y. Jun & T.Z. Mai: ADI method - Domain decomposition. Appl. Numer. Math. 56 (2006), 1092-1107. https://doi.org/10.1016/j.apnum.2005.09.008
  10. Y. Jun & T.Z. Mai: Rectangular domain decomposition method for parabolic problems. J. Korea Soc. Math. Educ. Ser. B: Pure and Appl. Math. 13 (2006), 281-294.
  11. Y. Jun: An efficient domain decomposition decomposition method for three-dimensional parabolic problems. Appl. Math. Comput. 215 (2009), 2815-2825. https://doi.org/10.1016/j.amc.2009.09.022
  12. Y. Jun: A stable noniterative Prediction/Correction domain decomposition method for hyperbolic problems. Appl. Math. Comput. 216 (2010), 2286-2292. https://doi.org/10.1016/j.amc.2010.03.064
  13. Y. Liu & H. Li: $H^{1}$-Galerkin mixed finite element methods for pseudo-hyperbolic equations. Appl. Math. Comput. 212 (2009), 446-457. https://doi.org/10.1016/j.amc.2009.02.039
  14. A. Mohebbi & M. Dehghan: High order compact solution of the one-space-dimensional linear hyperbolic equation. Numer. Methods Partial Differential Equations 24 (2008), 1222-1235. https://doi.org/10.1002/num.20313
  15. R.K. Mohanty: An unconditinally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficnents. Appl. Math. Comput. 165 (2005), 229-236. https://doi.org/10.1016/j.amc.2004.07.002
  16. R.K Mohanty: Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms. Appl. Math. Comput. 190 (2007), 1683-1690. https://doi.org/10.1016/j.amc.2007.02.097
  17. M. Ramezani, M. Dehghan & M. Razzaghi: Combined finite difference and spectral methods for the numerical solution of hyperbolic equation with an integral condition. Numer. Methods Partial Differential Equations 24 (2008), 1-8. https://doi.org/10.1002/num.20230
  18. K.K. Sharma & P. Singh: Hyperbolic partial differential-difference equation in the mathematical modeling of neuronal firing and its numerical solution. Appl. Math. Comput. 201 (2008), 229-238. https://doi.org/10.1016/j.amc.2007.12.051
  19. Z. Zhang & D. Deng: A new alternating-direction finite element method for hyperbolic equation. Numer. Methods Partial Differential Equations 23 (2007), 1530-1559. https://doi.org/10.1002/num.20240