A New Numerical Method for Solving Stiff Initial Value Problems

Bature Babangida,Musa H,Kane Il

doi:10.4172/2476-2296.1000136

Abstract

A new numerical method that computes 2–points simultaneously at each step of integration is derived. The numerical scheme is achieved by modifying an existing DI2BBDF method. The method is of order 2. The stability analysis of the new method indicates that it is both zero and A–stable, implying that it is suitable for stiff problems. The necessary and sufficient conditions for the convergence of the method are also established which proved the convergence of the method. Numerical results show that the method outperformed some existing algorithms in terms of accuracy.

Highlights

IntroductionConsider a system of first order stiff initial value problems (IVPs) of the form: y' =f (x, y) x ∈[a, b] y(x0 ) =y0
Consider a system of first order stiff initial value problems (IVPs) of the form: y' =f (x, y) x ∈[a, b] y(x0 ) =y0 (1)System (1) can be regarded as stiff if its exact solution contains very fast as well as very slow components [1]
It can be seen that the new method outperformed the existing 2-point diagonally implicit block backward differentiation formula in terms of accuracy

Summary

Introduction

Consider a system of first order stiff initial value problems (IVPs) of the form: y' =f (x, y) x ∈[a, b] y(x0 ) =y0. The solution is characterized by the presence of transient and steady state components, which restrict the step size of many numerical methods except methods with A-stability properties (Suleiman [2, 3]). This behaviour makes it difficult to develop suitable methods for solving stiff problems. Definition 3.2: A linear multistep method (LMM) is said to be A-stable if its stability region covers the entire negative half-plane. Consistency and zero stability are the necessary and sufficient conditions for the convergence of any numerical scheme. To give the visual impact on the performance of the new method, the graphs of Log (MAXE) against h for the problems tested are plotted in figures 2-4

Discussion

Numerical Results

Conclusion