L-stable and A-stable numerical method of order two for stiff differential equation

This article presents a finite difference method of order two for stiff differential equation that will overcome the effects of stiffness since it is A-stable. And, reflect the asymptotic behavior of the solution of the stiff problem since it is L-stable. The classical explicit Runge-Kutta methods of order two are not suitable for stiff problems since the numerical solution will not overcome the effects of stiffness and will not reflect the asymptotic behavior of the solution of the stiff problem (for example, Dahlquist problem). And, they are not both A-stable and L-stable. On applying Euler's implicit method, the numerical solution will overcome the effects of stiffness and will reflect the asymptotic behavior of the solution of the stiff problem. And hence, it is both A-stable and L-stable. But it is of order one method. A new numerical method which is both A-stable and L-stable and of order two for the numerical solution of a stiff differential equation is presented in this article. It will overcome the effects of stiffness and reflects the asymptotic behavior of the solution. The new method works well for large values of step size and works well in large domain. The new method is a modified form of the mid-point rule for the stiff problems. The rate of order of convergence is proved to be two both theoretically and numerically. Experimental results show the performance of the method based on the metrics such as stability function, stability region, order star fingers, the rate of order of theoretical and numerical convergence, absolute and relative errors, percentage of numerical solution accuracy and local and global truncation errors both numerically and graphically.