A Family of High Order One-Block Methods for the Solution of Stiff Initial Value Problems

Abstract

In this paper, we construct a family of high order self-starting one-block numerical methods for the solution of stiff initial value problems (IVP) in ordinary differential equations (ODE). The Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF) and Backward Differentiation Formulas (BDF) are used in the constructions. The E-transformation is applied to the triples and a family of self-starting methods are obtained. The family is for . The numerical implementation of the methods on some stiff initial value problems are reported to show the effectiveness of the methods. The computational rate of convergence tends to the theoretical order as h tends to zero.

How many k-step linear block methods exist and which of them is the most efficient and simplest one?

“…This 2-step block method has also appeared in [11][12][13] and [14] . Finally, in [15] Hongjiong and Bailin presented the 2-step block method that follows…”

“…in such a way that for every x i ∈ , the second-order spatial derivative appearing in (12) is approximated by means of the finite difference…”

A family of r-points 1-block implicit methods with optimized region of stability for stiff initial value problems in ordinary differential equations

“…This can be made possible by the application of suitable numerical methods. Researchers like [6,3,1,13,10,11,14,20] , etc have proposed different numerical methods for solving IVPs in ODEs but some of these methods are not self-starting methods and required single step methods to generate other starting values which is computationally cumbersome and prone to enormous error since the starting method and the method are not of the same order. Our interest in this work is to formulate a family of block methods with optimized region of stability suitable for stiff and non-stiff Initial Value problems in Ordinary Differential Equations of the form 𝑦 ′ = 𝑓(𝑥), 𝑦(𝑥 0 ) = 𝑦 0, 𝑥 ∈ [𝑎, 𝑏]…”

“…The proposed methods can also perform well for system of first order ODEs. Numerical block methods preserve the advantage of being self-starting methods and have minimal errors, it also has lesser time of computation since it solve simultaneously at all the grid points unlike some other numerical methods [11,5,15,1,7,18,19] etc. have worked on some numerical block methods for solution of (1) and have presented results that are competitive.…”

Construction of Stable High Order One-Block Methods Using Multi-Block Triple

“…In recent literatures, the focus is on block methods as a means of circumventing this barrier theorem. These methods are composed using different linear multistep formulas (LMF), (see [3,4,5,6,7,8,9]. In this paper, we show how one-block methods can be constructed using different multi-block methods as opposed to the known convention of using different single LMF.…”