Establishing the A(Α)-Stability of Extended Stable Parameter Dependent Nested Linear Multistep Methods with Off-Step Points for Stiff Ordinary Differential Equations

This study established the A(α)-stability of extended stable parameter dependent nested linear multistep method with off-step points of order seven for step number k ≤ 5 for the numerical integration of stiff and non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs). This method forms a super class of the linear multistep methods (LMMs), with properties that embedded the characteristics of LMM and hybrid methods. The method is hybrid in nature as a result of the incorporation of two or more off-step points and this makes this method a unique one. The primary aim of this method is to bypassing the Dahlquist order and stability barrier which serve as a set back to many works in linear multistep method. The stability properties of the methods were investigated and the intervals of absolute stability of the methods with step number k ≤ 5 are presented using the boundary locus techniques. The proposed method was found to attained a better stability property. The method is A- stable for k =1 and A(α) –stable for 2 ≤ k ≤ 5. The instability of the new methods set in when >5.which makes the methods more suitable for stiff initial value problems.

Nested multi-step methods, stiffly stable, boundary locus, A-stability, A(α)-stability, instability.

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