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