Convergence of the Fourth Order Variable Step Size Super Class of Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Abstract

In many fields of study such as science and engineering, various real life problems are created as mathematical models before they are solved. These models often lead to special class of ordinary differential equations known as stiff ODEs. A system is regarded as ‘stiff’; if the existing explicit numerical methods fail to efficiently integrate it, or when the step size is determined by the requirements of its stability, rather than the accurateness. The solution of stiff ODEs contains a component with both slowly and rapidly decaying rates due to a large difference in the time scale exhibited by the system. The stiffness property prevents the conventional explicit method from handling the problem efficiently. This nature of stiff ODEs has led to considerable research efforts in developing many implicit mathematical methods. This paper discussed the convergence and order of the current variable step size super-class of block backward differentiation formula (BBDF) for solving stiff initial value problems. The necessary conditions for the convergence of the fourth order variable step size super class of BBDF for solving stiff initial value problems, has been established in this work. It has been shown that the new method is both zero-stable and consistent, which are the requirements for the convergence of any numerical method. The order of the method is also derived to be four. It is therefore concluded that the method is convergent and has significance in solving more complex stiff initial value problems, and could be robustly applied in many fields of study.