Variable step block backward differentiation formula with independent parameter for solving stiff ordinary differential equations

Abstract

Over the last decade, the evolution of block backward differentiation formulas (BBDF) has involved the modifications of the formulation techniques in order to solve ordinary differential equations (ODEs). Better still, if the modified methods have the ability of computing solutions efficiently with any prescribed parameter. Therefore, this research focuses on the derivation of 2-point variable step block backward differentiation formulas (VSBBDF) that possesses independent parameter in the coefficients. In this formula, each block contains two points, which compute two approximate solutions simultaneously. Varying the value of parameter will lead to multiple choice of solutions with different level of accuracy. Since the method is derived using variable step size scheme, the strategy in controlling the step size ratio is also discussed. The capability of the derived method is demonstrated by solving initial value problem of stiff ODEs. A comparison of its performance with several existing methods is made to shed light on the superiority and shortcomings of VSBBDF with respect to independent parameter.

Variable Step Hybrid Block Method for the Approximation of Kepler Problem

“…The idea employed in their work is the combination of divided difference and Newton's interpolation formulas as the basis function in the design of the method. Other authors that derived variable step size methods include [17][18][19][20][21][22][23][24][25][26][27][28][29].…”

“…The scalar test to ascertain the zero-stability of a method was first proposed by [33]. Therefore, substituting y = f = λy (18) into the VSHBM at r = 1 gives…”