A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

References

• W.M. Abd-Elhameed, E.H. Doha, Y.H. Youssri, and M.A. Bassuony, New Tchebyshev-Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations, Numer. Meth. Part. D. E. 32 (2016), pp. 1553–1571. doi: 10.1002/num.22074
   Web of Science ®Google Scholar
• B. Chen, D. He, and K. Pan, A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers' equations, Numer. Math.-Theory Me. 11 (2018), pp. 299–320.
   Google Scholar
• P. C. Chu and C. Fan, A three-point combined compact difference scheme, J. Comput. Phys. 140 (1998), pp. 370–399. doi: 10.1006/jcph.1998.5899
   Web of Science ®Google Scholar
• M. Dehghan and A. Mohebbi, High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation, Numer. Meth. Part. D. E. 25 (2009), pp. 232–243. doi: 10.1002/num.20341
   Web of Science ®Google Scholar
• M. Dehghan and A. Shokri, A numerical method for solving the hyperbolic telegraph equation, Numer. Meth. Part. D. E. 24 (2008), pp. 1080–1093. doi: 10.1002/num.20306
   Web of Science ®Google Scholar
• M. Dehghan and A. Shokri, A meshless method for numerical solution of a linear hyperbolic equation with variable coefficients in two space dimensions, Numer. Meth. Part. D. E. 25 (2009), pp. 494–506. doi: 10.1002/num.20357
   Web of Science ®Google Scholar
• H. Ding and Y. Zhang, A new fourth-order compact finite difference scheme for the two-dimensional second-order hyperbolic equation, J. Comp. Appl. Math. 230 (2009), pp. 626–632. doi: 10.1016/j.cam.2009.01.001
   Web of Science ®Google Scholar
• G. Gao and H. Sun, Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations, Commun. Comput. Phys. 17 (2015), pp. 487–509. doi: 10.4208/cicp.180314.010914a
   Web of Science ®Google Scholar
• G. Gao and H. Sun, Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions, J. Comput. Phys. 298 (2015), pp. 520–538. doi: 10.1016/j.jcp.2015.05.052
   Web of Science ®Google Scholar
• D. He, An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear hyperbolic equation, Numer. Algorithms. 72 (2016), pp. 1103–1117. doi: 10.1007/s11075-015-0082-7
   Web of Science ®Google Scholar
• D. He and K. Pan, An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrödinger equations with variable coefficients in two and three dimensions, Comput. Math. Appl. 73 (2017), pp. 2360–2374. doi: 10.1016/j.camwa.2017.04.009
   Web of Science ®Google Scholar
• D. He and K. Pan, A fifth-order combined compact difference scheme for Stokes flow on polar geometries, E. Asian J. Appl. Math. 7 (2018), pp. 714–727. doi: 10.4208/eajam.200816.300517a
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• S. Lee, J. Liu, and H. Sun, Combined compact difference scheme for linear second-order partial differential equations with mixed derivative, J. Comput. Appl. Math. 264 (2014), pp. 23–37. doi: 10.1016/j.cam.2014.01.004
   Web of Science ®Google Scholar
• L. Li, H. Sun, and S. Tam, A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schro¨dinger equations, Comput. Phys. Commun. 187 (2015), pp. 38–48. doi: 10.1016/j.cpc.2014.10.008
   Google Scholar
• M. Li, Z. Zheng, and K. Pan, An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection-diffusion equations, Adv. diff. Equ. 2018 (2018), p. 178. doi: 10.1186/s13662-018-1631-x
   Google Scholar
• J. Liu and K. Tang, A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation, Int. J. Comput. Math. 87 (2010), pp. 2259–2267. doi: 10.1080/00207160802624133
   Web of Science ®Google Scholar
• K. Mahesh, A family of high order finite difference schemes with good spectral resolution, J. Comput. Phys. 145 (1998), pp. 332–358. doi: 10.1006/jcph.1998.6022
   Web of Science ®Google Scholar
• A. Mardani, M.R. Hooshmandasl, M.M. Hosseini, and M.H. Heydari, Moving least squares (MLS) method for the nonlinear hyperbolic telegraph equation with variable coefficients, Int. J. Comp. Meth.-Sing. 14 (2017), p. 1750026.
   Web of Science ®Google Scholar
• 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, Appl. Math. Comput. 152 (2004), pp. 799–806.
   Web of Science ®Google Scholar
• R.K. Mohanty, An unconditionally stable difference scheme for the one-space-dimensional linear hyperbolic equation, Appl. Math. Lett. 17 (2004), pp. 101–105. doi: 10.1016/S0893-9659(04)90019-5
   Web of Science ®Google Scholar
• R.K. Mohanty, An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput. 165 (2005), pp. 229–236.
   Web of Science ®Google Scholar
• R.K. Mohanty, New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations, Int. J. Comput. Math. 86 (2009), pp. 2061–2071. doi: 10.1080/00207160801965271
   Web of Science ®Google Scholar
• R.K. Mohanty, M.K. Jain, and K. George, On the use of high order difference methods for the system of one space second order non-linear hyperbolic equations with variable coefficients, J. Comput. Appl. Math. 72 (1996), pp. 421–431. doi: 10.1016/0377-0427(96)00011-8
   Web of Science ®Google Scholar
• R.K. Mohanty, M.K. Jain, and U. Arora, An unconditionally stable ADI method for the linear hyperbolic equation in three space dimensional, Int. J. Comput. Math. 79 (2002), pp. 133–142. doi: 10.1080/00207160211918
   Web of Science ®Google Scholar
• R.K. Mohanty and M.K. Jam, An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation, Numer. Meth. Part. D. E. 17 (2001), pp. 684–688. doi: 10.1002/num.1034
   Web of Science ®Google Scholar
• A. Mohebbi and M. Dehghan, High order compact solution of the one-space-dimensional linear hyperbolic equation, Numer. Meth. Part. D. E. 24 (2008), pp. 1222–1235. doi: 10.1002/num.20313
   Web of Science ®Google Scholar
• T. Nihei and K. Ishii, A fast solver of the shallow water equations on a sphere using a combined compact difference scheme, J. Comput. Phys. 187 (2003), pp. 639–659. doi: 10.1016/S0021-9991(03)00152-9
   Web of Science ®Google Scholar
• K. Pan, D. He, and H. Hu, An extrapolation cascadic multigrid method combined with a fourth-order compact scheme for 3D Poisson equation, J. Sci. Comput. 70 (2017), pp. 1180–1203. doi: 10.1007/s10915-016-0275-9
   Web of Science ®Google Scholar
• S. Pandit, M. Kumar, and S. Tiwari, Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients, Comput. Phys. Commun. 187 (2015), pp. 83–90. doi: 10.1016/j.cpc.2014.10.013
   Web of Science ®Google Scholar
• D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1955), pp. 28–41. doi: 10.1137/0103003
   Google Scholar
• J. Rashidinia, R. Mohammadi, and R. Jalilian, Spline methods for the solution of hyperbolic equation with variable coefficients, Numer. Meth. Part. D. E. 32 (2006), pp. 1–9.
   Google Scholar
• T.K. Sengupta, V. Lakshmanan, and V. Vijay, A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys. 228 (2009), pp. 3048–3071. doi: 10.1016/j.jcp.2009.01.003
   Web of Science ®Google Scholar
• T.K. Sengupta, V. Vijay, and S. Bhaumik, Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties, J. Comput. Phys. 228 (2009), pp. 6150–6168. doi: 10.1016/j.jcp.2009.05.038
   Web of Science ®Google Scholar
• H. Sun and L. Li, A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Commun. 185 (2014), pp. 790–797. doi: 10.1016/j.cpc.2013.11.009
   Web of Science ®Google Scholar
• Q. Wang, K. Pan, and H. Hu, Unique solvability of the CCD scheme for convection-diffusion equations with variable convection coefficients, Adv. Diff. Equ. 2018 (2018), p. 163. doi: 10.1186/s13662-018-1591-1
   Google Scholar
• J. Zhang and J. Zhao, Truncation error and oscillation property of the combined compact difference scheme, Appl. Math. Comput. 161 (2005), pp. 241–251.
   Web of Science ®Google Scholar