Accuracy Improvement of Block Backward Differentiation Formulas for Solving Stiff Ordinary Differential Equations Using Modified Versions of Euler's Method

Abstract

In this study, the fully implicit 2-point block backward differentiation formulas (BBDF) method has been successfully utilized for solving stiff ordinary differential equations (ODEs) by taking into account the uses of new starting methods namely, modified Euler's method (MEM), improved modified Euler's method (IMEM), and new Euler's method (NEM). The reason of proposing the BBDF is that the method has been proven useful for stiff ODEs due to its A-stable properties. Furthermore, the method is able to approximate the solutions at two points simultaneously at each step. The proposed method is also implemented through Newton's iteration procedure, which involves the calculation of the Jacobian matrix. Accuracy of the method is evaluated based on its performance in solving linear and non-linear initial value problems (IVPs) of first order stiff ODEs with transient and steady-state solutions. Some comparisons are made with the conventional BBDF approach for indicating the reliability of the proposed method. Numerical results indicate that not only classical Euler's method provides accurate solutions for BBDF, but also the numerous modified versions of Euler's methods improve the accuracy of BBDF, in terms of absolute error at certain step size and stage of iteration.