Explicit and not fully implicit optimal and uniform finite difference schemes of order one for stiff initial value  problems

K. Selvakumar
Department of   Mathematics , Anna University of Technology  Tirunelveli,Tirunelveli-627 007,  Tamil Nadu, India.


This paper presents explicit and not fully implicit finite difference schemes of order one for stiff initial value problems with a small parameter  multiplying the first derivative.   The schemes are modified form of  classical Euler’s rule of order one. And the schemes are  both uniform and optimal  with respect to  the small parameter , that is,  the solution of  the difference scheme  satisfies the  error estimates  of the form:

| u (  ti ) - ui | C min ( h ,   )

where C is independent of   i, h and . Here  h is the mesh size  and ti   is any mesh point.  The explicit scheme  presented in this paper solves  the open problem proposed by  Doolan et al., [6]. The open problem is; “Is it possible to obtain optimal  or quasi-optimal  methods which are not fully implicit? “. The  implicit  scheme presented in this paper which is not fully implicit  is also a solution for the  open problem. Finally numerical  experiments  are presented.

Keywords:   initial layer , stiff initial value problems, singular perturbation problems,   exponentially fitted, uniformly convergent, asymptotic  expansion,  finite difference schemes.

AMS (MOS)  subject classification:  65F05, 65N30, 65N35, 650Y05.




International eJournal of Mathematical Sciences, Technology and Humanities

Volume 2, Issue 3, Pages:  561 - 576