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

Diserahkan: 3 Februari 2012 / Diterima: 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

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*Pengarang untuk surat-menyurat; email: masn@tmsk.uitm.edu.my