Construction of L(α)-stable one-step Fourth-Order Third-Derivative Parameter-Dependent General Linear Method for Stiff ODEs

A one-step third-derivative parameter-dependent general linear method is proposed for the numerical solution of stiff ordinary differential equations. The method is constructed within the general linear methods framework and incorporates solution derivatives up to the third order, providing additional flexibility for accuracy and stability control. An implicit formulation is adopted and the free parameters are determined by enforcing consistency and fourth-order accuracy conditions. Stability analysis based on the linear test equation shows that the resulting rational stability function satisfies the A-stability property and, in addition, achieves L(α)-stability, as the stability function vanishes for large negative values of the stiffness parameter. This ensures strong damping of stiff components and suppresses nonphysical oscillations. The proposed scheme is a one-step, fourth-order, L(α)-stable method that combines high accuracy with excellent stability characteristics, making it suitable for stiff initial value problems.