2-Point Block BDF Method with Off-Step Points for Solving Stiff ODEs

Abstract

In this paper, 2-point block method with two off-step points based on Backward Differentiation Formula (BDF) for solving stiff ODEs is formulated. The strategy of the developed method is to calculate two solution values of the method with two off-step points simultaneously at each iteration. Stability region and convergence of the method are also generated. The numerical results obtained are compared with the fifth order 2-point block BDF method to compare the enhancement of the method in terms of accuracy.

A New Numerical Method for Solving Stiff Initial Value Problems

“…This behaviour makes it difficult to develop suitable methods for solving stiff problems. However, efforts have been made by researchers, such as Abasi [4], Alt [5], Alvarez [6], Cash [7], Dahlquist [1], Ibrahim [8][9][10], Musa [11][12][13][14], Suleiman [2,3], Yatim [15] and Zawawi [16] among others, to develop methods for stiff ODEs. The need to obtain an efficient numerical approximation in terms of accuracy and computational time have attracted some researchers such as Alexander [17] with diagonally implicit Runge-Kutta for stiff ODEs, Ababneh [18] with design of new diagonally implicit Runge-Kutta for stiff problems, Ismail [19] with embedded pair of diagonally implicit Runge-Kutta for solving ODEs, Zawawi [20] with diagonally implicit block backward differentiation formulas for solving ODEs.…”

Stability region of two-point variable step–block backward differentiation formulae

“…Furthermore, there are variety of solver, which are based on BBDF method that are available to solve stiff ODE developed in literature. [3][4][5][6][7][8][9][10] Basically, for method to be of practical importance, it must have a region of absolute stability to ensure that the method will be able to solve at least for the mildly stiff problems. 11 In this article, we are interested to analyse the stability of VS-BBDF method with constant step size, half the step size and increment the step size by a factor 1.8.…”

Implicit Block Hybrid-Like Method for Solving System of First Order Ordinary Differential Equations

“…Akinfenwa et al [3] implemented the proposed method as the self starting method that does not require the starting values to proceed while James et al [10] implemented the proposed method in predictorcorrector mode. Abasi et al [1] derived the block hybrid method based on backward differentiation formula and interpolation polynomial that interpolates the value of at two main points and two off-step points.…”

“…Here, we are going to apply the block hybrid-like method in predictor-corrector mode to solve system of first order ODEs (1). The implementation of this method approximates the solution of y at the main point and the off-step point simultaneously.…”