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Section B

Compact alternating direction implicit method to solve two-dimensional nonlinear delay hyperbolic differential equations

, &
Pages 964-982 | Received 02 Dec 2012, Accepted 26 May 2013, Published online: 01 Jul 2013

References

  • M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Anal. 54 (2003), pp. 1469–1491. doi: 10.1016/S0362-546X(03)00197-4
  • Y.R. Chi, A high-precision compact difference scheme for a class of nonlinear delay parabolic partial differential equations, J. Yanbian Univ. Nat. Sci. 36(4) (2010), pp. 287–290.
  • P.C. Chu and C.W. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), pp. 370–399. doi: 10.1006/jcph.1998.5899
  • B.T. Cui, Y.Q. Liu, and F.Q. Deng, Some oscillation problems for impulsive hyperbolic differential systems with several delays, Appl. Math. Comput. 146 (2003), pp. 667–679. doi: 10.1016/S0096-3003(02)00611-2
  • D.W. Deng and Z.Z. Zhang, A new high-order algorithm for a class of nonlinear evolution equation, J. Phys. A 41 (2008), pp. 1–17.
  • D.W. Deng and C.J. Zhang, Application of a fourth-order compact ADI method to solve a two-dimensional linear hyperbolic equation, Int. J. Comput. Math. 90(2) (2013), pp. 273–291. doi: 10.1080/00207160.2012.713475
  • H.F. Ding and Y.X. Zhang, A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation, J. Comput. Appl. Math. 230 (2009), pp. 626–632. doi: 10.1016/j.cam.2009.01.001
  • J. DouglasJr., On the numerical integration of uxx + uyy=ut by implicit methods, J. Soc. Ind. Appl. Math. 3 (1955), pp. 42–65. doi: 10.1137/0103004
  • J. DouglasJr. and J.E. Gunn, A general formulation of alternating direction method. Part I: Parabolic and hyperbolic problems, Numer. Math. 6 (1964), pp. 428–453. doi: 10.1007/BF01386093
  • L.L. Du, W. Fu, and M.S. Fan, Oscillatory solutions of delay hyperbolic system with distributed deviating arguments, Appl. Math. Comput. 154 (2004), pp. 521–529. doi: 10.1016/S0096-3003(03)00732-X
  • E.G. D'Yakonov, Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed partial derivatives, Zh. Vychisl. Mat. Mat. Fiz. 4 (1964), pp. 935–941.
  • E. Fridmana and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica 45 (2009), pp. 194–201. doi: 10.1016/j.automatica.2008.06.006
  • X.L. Fu and L.Q. Zhang, Forced oscillation for impulsive hyperbolic boundary value problems with delay, Appl. Math. Comput. 158 (2004), pp. 761–780. doi: 10.1016/j.amc.2003.08.148
  • Z. Gao and S.S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations, Appl. Numer. Math. 61 (2011), pp. 593–614. doi: 10.1016/j.apnum.2010.12.004
  • P.J.V. Houwen, B.P. Sommeijer, and C.T.H. Baker, On the stability of predictor-corrector methods for parabolic equations with delay, IMA J. Numer. Anal. 6 (1986), pp. 1–23. doi: 10.1093/imanum/6.1.1
  • C.M. Huang and S. Vandewalle, An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays, SIAM J. Sci. Comput. 25(5) (2004), pp. 1608–1632. doi: 10.1137/S1064827502409717
  • C.M. Huang and S. Vandewalle, Unconditionally stable difference methods for delay partial differential equations, Numer. Math. 122(3) (2012), pp. 579–601. doi: 10.1007/s00211-012-0467-7
  • S. Karaa, A high-order compact ADI method for solving three-dimensional unsteady convection-diffusion problems, Numer. Methods Partial Differential Equations 22 (2006), pp. 983–993. doi: 10.1002/num.20134
  • S. Karaa, A high-order ADI method for parabolic problems with variable coefficients, Int. J. Comput. Math. 86 (2009), 109–120. doi: 10.1080/00207160802217227
  • S. Karaa, Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations, Int. J. Comput. Math. 87 (2010), pp. 3030–3038. doi: 10.1080/00207160902878548
  • S. Karaa and J. Zhang, High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys. 198 (2004), pp. 1–9. doi: 10.1016/j.jcp.2004.01.002
  • K. Kreith, T. Kusano, and N. Yoshida, Oscillation properties of nonlinear hyperbolic equations, SIAM J. Math. Anal. 15 (1984), pp. 570–578. doi: 10.1137/0515043
  • D.F. Li, C.J Zhang, and W.S. Wang, Long time behavior of non-Fickian delay reaction-diffusion equations, Nonlinear Anal. Real World Appl. 13 (2012), pp. 1401–1415. doi: 10.1016/j.nonrwa.2011.11.005
  • H.L. Liao, Z.Z. Sun, and H.S. Shi, Error estimate of fourth-order compact scheme for linear schrödinger equations, SIAM J. Numer. Anal. 47(6) (2010), pp. 4381–4401. doi: 10.1137/080714907
  • R.K. Mohanty, An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation, Appl. Math. Lett. 17(1) (2004), pp. 101–105. doi: 10.1016/S0893-9659(04)90019-5
  • R.K. Mohanty and M.K. Jain, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Int. J. Comput. Math. 79 (2002), pp. 133–142. doi: 10.1080/00207160211918
  • D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic equations, J. Soc. Ind. Appl. Math. 3 (1955), pp. 28–41. doi: 10.1137/0103003
  • L.Z. Qian and H.B. Gu, High order compact scheme combined with extrapolation technique for solving convection-diffusion equations, J. Shandong Univ. Nat. Sci. 46(12) (2011), pp. 39–43.
  • A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, New York, 1997.
  • A.V. Rezounenko and J.H. Wu, A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors, J. Comp. Appl. Math. 190 (2006), pp. 99–113. doi: 10.1016/j.cam.2005.01.047
  • F. Rodríguez, M. Roales, and J.A. Martín, Exact solutions and numerical approximations of mixed problems for the wave equation with delay, Appl. Math. Comput. 219 (2012), pp. 3178–3186. doi: 10.1016/j.amc.2012.09.050
  • Z.Z. Sun, The Numerical Methods for Partial Differential Equations, Science Press, Beijing, 2005 (in Chinese).
  • Z.Z. Sun, A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Appl. Math. Model. 37 (2013), pp. 742–752. doi: 10.1016/j.apm.2012.02.036
  • H.W. Sun and J. Zhang, A high order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numer. Methods Partial Differential Equations 20 (2004), pp. 18–32. doi: 10.1002/num.10075
  • Z.F. Tian and Y.B. Ge, A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math. 198 (2007), pp. 268–286. doi: 10.1016/j.cam.2005.12.005
  • P.K.C. Wang, Asymptotic stability of a time-delayed diffusion system, J. Appl. Mech. Ser. E. 30 (1963), pp. 500–504. doi: 10.1115/1.3636609
  • P.G. Wang, Forced oscillation of a class of delay hyperbolic equation boundary value problem, Appl. Math. Comput, 103 (1999), pp. 15–25. doi: 10.1016/S0096-3003(98)10061-9
  • Y.M. Wang, Error and extrapolation of a compact LOD method for parabolic differential equations, J. Comput. Appl. Math. 235 (2011), pp. 1367–1382. doi: 10.1016/j.cam.2010.08.024
  • Y. Wang and J. Zhang, Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation, J. Comput. Phys. 228 (2009), pp. 137–146. doi: 10.1016/j.jcp.2008.09.002
  • J.H. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
  • Y. Zhang, The stability of linear partial differential systems with time delay, Int. J. Systems Sci. 26 (1995), pp. 1747–754. doi: 10.1080/00207729508929035
  • Q.F. Zhang and C.J. Zhang, A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations, Appl. Math. Lett. 26 (2013), pp. 306–312. doi: 10.1016/j.aml.2012.09.015
  • Q.F. Zhang and C.J. Zhang, A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay, Commun. Nonlinear Sci. Numer. Simul. (2013), doi: http://dx.doi.org/10.1016/j.cnsns.2013.05.018.

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