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
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