Sains Malaysiana 42(3)(2013): 341–346

A Fourth-order Compact Finite Difference Scheme for the Goursat Problem

(Skema Beza Terhingga Padat Peringkat Empat untuk Masalah Goursat)

Mohd Agos Salim bin Nasir*

Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Malaysia

40450 Shah Alam, Selangor D.E., Malaysia

Ahmad Izani bin Md Ismail

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia

Received: 3 February 2012/Accepted: 17 September 2012

ABSTRACT

A high-order uniform Cartesian grid compact finite difference scheme for the Goursat problem is developed. The basic idea of high-order compact schemes is to find the compact approximations to the derivatives terms by differentiating centrally the governing equations. Our compact scheme will approximate the derivative terms by involving the higher terms and reducing the number of grid points. The compact finite difference scheme is given for general form of the Goursat problem in uniform domain and illustrates the performance by applying a linear problem. Numerical experiments have been conducted with the new scheme and encouraging results have been obtained. In this paper we present the compact finite difference scheme for the Goursat problem. With the aid of computational software the scheme was programmed for determining the relative errors of linear Goursat problem.

Keywords: Compact finite difference; consistency; convergence; Goursat problem; stability

ABSTRAK

Skema beza terhingga padat bagi grid Kartesan seragam peringkat tinggi untuk masalah Goursat dibincangkan. Idea asas bagi skema padat peringkat tinggi ialah untuk mendapatkan penghampiran padat sebutan-sebutan terbitan dengan membezakan secara memusat persamaan yang bersekutu. Skema padat kami akan membuat penghampiran sebutan-sebutan terbitan dengan melibatkan sebutan-sebutan peringkat lebih tinggi dan mengurangkan bilangan titik-titik grid. Skema beza terhingga padat diberikan dalam bentuk am untuk masalah Goursat di dalam domain seragam dan prestasinya digambarkan dengan mengaplikasi satu masalah linear. Uji kaji berangka dengan skema baru telah dijalankan dan keputusan memberangsangkan telah  diperoleh. Dalam kertas  ini kami berikan skema beza terhingga padat untuk masalah Goursat. Dengan bantuan perisian pengkomputeran, skema telah diatur cara untuk menentukan pelbagai ralat relatif bagi masalah Goursat linear.

Kata kunci: Beza terhingga padat; ketekalan; kestabilan; masalah Goursat; penumpuan

REFERENCES

Ahmed, M.O. 1997. An exploration of compact finite difference methods for the numerical solution of PDE. PhD dissertation. Western Ontario London University (unpublished).

Boersma, B.J. 2005. A staggered compact finite difference formulation for the compressible Navier-Stokes equations. Journal of Computational Physics 208: 675-690.

Bodenmann, R. & Schroll, H.J. 1996. Compact difference methods applied to initial boundary value problems for mixed systems. Numer. Math. 73: 291-309.

Day, J.T. 1966. A Runge-Kutta method for the numerical solution of the Goursat problem in hyperbolic partial differential equations. Computer Journal 9: 81-83.

Evans, D.J. & Sanugi, B.B. 1988. Numerical solution of the Goursat problem by a nonlinear trapezoidal formula. Applied Mathematics Letter 3: 221-223.

Fletcher, C.A.J. 1990. Computational Techniques for Fluid Dynamics. 2^nd ed. Berlin: Springer Verlag.

Frisch, R.A. & Cheo, B.R. 1972. On a bounded one dimensional Poisson-Vlasov system. Society for Industry and Applied Mathematics 3: 362-368.

Garabedian, P. 1964. Partial Differential Equations. New York: John Wiley & Sons Inc.

Hillion, P. 1992. A note on the derivation of paraxial equation in nonhomogeneous media. SIAM Journal of Applied Mathematics 2: 337-346.

Kyei, Y. 2004. 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, USA (unpublished).

Kaup, D.J. & Newell, A.C. 1978. The Goursat and Cauchy problems for the Sine Gordon equation. SIAM Journal of Applied Mathematics 1: 37-54.

Li, J. & Visbal, M.R. 2006. High order compact schemes for nonlinear dispersive waves. Journal of Scientific Computing 26: 1-23.

Li, J. & Chen, Y. 2004. High order compact schemes for dispersive media. Electronics Letters 40: 14.

Li, M. & Tang, T. 2001. A compact fourth order finite difference scheme for unsteady viscous incompressible flows. Journal of Scientific Computing 16: 29-45.

Mclaughlin, J.R., Polyakov, P.L. & Sacks, P.E. 1994. Reconstruction of a spherical symmetric speed of sound. SIAM Journal of Applied Mathematics 5: 1203-1223.

Nasir, M.A.S. & Ismail, A.I.M. 2005. A new finite difference scheme for the Goursat problem. IASME Transactions 2: 98-103.

Nasir, M.A.S. & Ismail, A.I.M. 2004. Numerical solution of a linear Goursat problem: Stability, consistency and convergence. WSEAS Transactions on Mathematics 3: 616-620.

Richtmyer, R. & Morton, K.W. 1967. Difference Methods for Initial Value Problems. New York: John Wiley & Sons.

Spotz, W.F. 1995. High order compact finite difference schemes for computational mechanics. PhD dissertation. Texas University. Austin, USA (unpublished).

Twizell, E.H. 1984. Computational Methods for Partial Differential Equations. Chichester: Ellis Horwood Limited.

Wazwaz, A.M. 1995. The decomposition method for approximate solution of the Goursat problem. Applied Mathematics and Computation 69: 299-311.

Wazwaz, A.M. 1993. On the numerical solution for the Goursat problem. Applied Mathematics and Computation 59: 89-95.

Zhao, J., Zhang, T. & Corless, R.M. 2006. Convergence of the compact finite difference method for second order elliptic equations. Applied Mathematics and Computation 182: 1454-1469.

Zhao, J., Corless, R.M. & Davison, M. 2005. Financial applications of symbolically generated compact finite difference formulae. Dept. of Applied Mathematics. London: Western Ontario London University.

*Corresponding author; email: masn@tmsk.uitm.edu.my