Applications of High-Order Compact Finite Difference Scheme to Nonlinear Goursat Problems

Authors: Mohd Agos Salim Nasir, Ahmad Izani Md. Ismail

Abstract:

Several numerical schemes utilizing central difference approximations have been developed to solve the Goursat problem. However, in a recent years compact discretization methods which leads to high-order finite difference schemes have been used since it is capable of achieving better accuracy as well as preserving certain features of the equation e.g. linearity. The basic idea of the new scheme is to find the compact approximations to the derivative terms by differentiating centrally the governing equations. Our primary interest is to study the performance of the new scheme when applied to two Goursat partial differential equations against the traditional finite difference scheme.

Keywords: Goursat problem, partial differential equation, finite difference scheme, compact finite difference

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1080734

References:

[1] J. Zhao, T. Zhang and R.M. Corless. Convergence of the compact finite difference method for second order elliptic equations. Applied Mathematics and Computation. 2006, pg. 1454-1469.
[2] J. Li and M.R. Visbal. High order compact schemes for nonlinear dispersive waves. Journal of scientific computing. 2006, pg. 1-23.
[3] B.J. Boersma. A staggered compact finite difference formulation for the compressible Navier-Stokes equations. Journal of Computational Physics. 2005, pg. 675-690.
[4] J. Zhao, R.M. Corless and M. Davison. Financial applications of symbolically generated compact finite difference formulae. Dept. of Applied Mathematics. Western Ontario London University. 2005.
[5] J. Li and Y. Chen. High order compact schemes for dispersive media. Electronics Letters. 2004, pg. 14.
[6] Y. Kyei. Higher order Cartesian grid based finite difference methods for elliptic equations on irregular domains and interface problems and their applications. PhD dissertation. North Carolina State University. 2004.
[7] M.A.S. Nasir and A.I.M. Ismail. Numerical solution of a linear Goursat problem: Stability, consistency and convergence. WSEAS Transactions on Mathematics. 2004.
[8] M. Li and T. Tang. A compact fourth order finite difference scheme for unsteady viscous incompressible flows. Journal of Scientific Computing. 2001, pg. 29-45.
[9] M.O. Ahmed. An exploration of compact finite difference methods for the numerical solution of PDE. PhD dissertation. Western Ontario London University. 1997.
[10] R. Bodenmann and H.J. Schroll. Compact difference methods applied to initial boundary value problems for mixed systems. Numer. Math. 1996, pg. 291-309.
[11] A.M. Wazwaz. The decomposition method for approximate solution of the Goursat problem. Applied Mathematics and Computation. 1995, pg. 299-311.
[12] W.F. Spotz. High order compact finite difference schemes for computational mechanics. PhD dissertation. Texas University. 1995.
[13] J.R. Mclaughlin, P.L. Polyakov and P.E. Sacks. Reconstruction of a spherical symmetric speed of sound. SIAM Journal of Applied Mathematics. 1994, pg. 1203-1223.
[14] A.M. Wazwaz. On the numerical solution for the Goursat problem. Applied Mathematics and Computation. 1993, pg. 89-95.
[15] P. Hillion. A note on the derivation of paraxial equation in nonhomogeneous media. SIAM Journal of Applied Mathematics. 1992, pg. 337-346.
[16] D.J. Evans and B.B. Sanugi. Numerical solution of the Goursat problem by a nonlinear trapezoidal formula. Applied Mathematics Letter. 1988, pg. 221-223.
[17] E.H. Twizell. Computational Methods For Partial Differential Equations. Ellis Horwood Limited. 1984.
[18] D.J. Kaup and A.C. Newell. The Goursat and Cauchy problems for the Sine Gordon equation. SIAM Journal of Applied Mathematics. 1978, pg. 37-54.
[19] T.Y. Cheung. Three Nonlinear Initial Value Problems of The Hyperbolic Type. SIAM J. Numer. Anal. 1977, pg. 484-491.
[20] R.A. Frisch and B.R. Cheo. On a bounded one dimensional Poisson- Vlasov system. Society for Industry and Applied Mathematics. 1972.
[21] J.T. Day. A Runge-Kutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations. Computer Journal. 1966, pg. 81-83.