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

In this article, a high-order compact alternating direction implicit method combined with a Richardson extrapolation technique is developed to solve a class of two-dimensional nonlinear delay hyperbolic differential equations. The solvability, stability and convergence of the method are analysed simultaneously in L2- and H1-norms by the discrete energy method. Numerical experiments are provided to demonstrate the accuracy and efficiency of the schemes.

[1]  Samir Karaa,et al.  Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations , 2010, Int. J. Comput. Math..

[2]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[3]  Samir Karaa.  A high-order compact ADI method for solving three-dimensional unsteady convection-diffusion problems , 2006 .

[4]  Feiqi Deng,et al.  Some oscillation problems for impulsive hyperbolic differential systems with several delays , 2003, Appl. Math. Comput..

[5]  Lili Du,et al.  Oscillatory solutions of delay hyperbolic system with distributed deviating arguments , 2004, Appl. Math. Comput..

[6]  P. Wang,et al.  Asymptotic Stability of a Time-Delayed Diffusion System , 1963 .

[7]  Chengjian Zhang,et al.  Application of a fourth-order compact ADI method to solve a two-dimensional linear hyperbolic equation , 2013, Int. J. Comput. Math..

[8]  Jun Zhang,et al.  High order ADI method for solving unsteady convection-diffusion problems , 2004 .

[9]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[10]  Y. Ge,et al.  A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems , 2007 .

[11]  Chengjian Zhang,et al.  Long time behavior of non-Fickian delay reaction–diffusion equations ☆ , 2012 .

[12]  Peiguang Wang.  Forced oscillation of a class of delay hyperbolic equation boundary value problem , 1999, Appl. Math. Comput..

[13]  Xilin Fu,et al.  Forced oscillation for impulsive hyperbolic boundary value problems with delay , 2004, Appl. Math. Comput..

[14]  Zhiyue Zhang,et al.  A new high-order algorithm for a class of nonlinear evolution equation , 2008 .

[15]  Samir Karaa,et al.  A high-order ADI method for parabolic problems with variable coefficients , 2009, Int. J. Comput. Math..

[16]  Shusen Xie,et al.  Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations , 2011 .

[17]  Zhi-Zhong Sun,et al.  Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations , 2010, SIAM J. Numer. Anal..

[18]  Arnold W. Heemink,et al.  Performance robustness analysis of Kalman filter for linear discrete-time systems under plant and noise uncertainty , 1995 .

[19]  Zhi-zhong Sun,et al.  A linearized compact difference scheme for a class of nonlinear delay partial differential equations , 2013 .

[20]  Marcel Bauer,et al.  Numerical Methods for Partial Differential Equations , 1994 .

[21]  R. K. Mohanty.  An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation , 2004, Appl. Math. Lett..

[22]  R. K. Mohanty,et al.  An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation , 2001 .

[23]  Chengjian Zhang,et al.  A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay , 2013, Commun. Nonlinear Sci. Numer. Simul..

[24]  Ben P. Sommeijer,et al.  On the Stability of Predictor-Corrector Methods for Parabolic Equations with Delay , 1984 .

[25]  Yin Wang,et al.  Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation , 2009, J. Comput. Phys..

[26]  J. J. Douglas.  On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods , 1955 .

[27]  Stefan Vandewalle,et al.  An Analysis of Delay-Dependent Stability for Ordinary and Partial Differential Equations with Fixed and Distributed Delays , 2004, SIAM J. Sci. Comput..

[28]  Hengfei Ding,et al.  A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation , 2009 .

[29]  Emilia Fridman,et al.  Exponential stability of linear distributed parameter systems with time-varying delays , 2009, Autom..

[30]  Hai-wei Sun,et al.  A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations , 2004 .

[31]  Chengjian Zhang,et al.  A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations , 2013, Appl. Math. Lett..

[32]  Stefan Vandewalle,et al.  Unconditionally stable difference methods for delay partial differential equations , 2012, Numerische Mathematik.

[33]  P. Chu,et al.  A Three-Point Combined Compact Difference Scheme , 1998 .

[34]  N. Yoshida,et al.  Oscillation Properties of Nonlinear Hyperbolic Equations , 1984 .

[35]  Jianhong Wu,et al.  A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors , 2006 .

[36]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[37]  J. Douglas,et al.  A general formulation of alternating direction methods , 1964 .

[38]  Fabien Crauste,et al.  Global stability of a partial differential equation with distributed delay due to cellular replication , 2003, 0904.2472.

[39]  Yuan-Ming Wang,et al.  Error and extrapolation of a compact LOD method for parabolic differential equations , 2011, J. Comput. Appl. Math..

[40]  Francisco Rodríguez-Mateos,et al.  Exact solutions and numerical approximations of mixed problems for the wave equation with delay , 2012, Appl. Math. Comput..

[41]  R. K. Mohanty,et al.  An Unconditionally Stable ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions , 2002, Int. J. Comput. Math..