A class of $A(\alpha)$-stable numerical methods for stiff problems in ordinary differential equations

The $A(\alpha)$-stable numerical methods (ANM) for the number of steps $k\le7$ for stiff initial value problems (IVPs) in ordinary differential equations (ODEs) are proposed. The discrete schemes proposed from their equivalent continuous schemes are obtained. The scaled time variable $t$ in a continuous method, which determines the discrete coefficients of the discrete method is chosen in such a way as to ensure that the discrete scheme attains a high order and $A(\alpha)$-stability. We select the value of $\alpha$ for which the schemes proposed are absolutely stable. The new algorithms are found to have a comparable accuracy with that of the backward differentiation formula (BDF) discussed in [12] which implements the Ode15s in the Matlab suite.