A New Class of General Linear Method with Inherent Quadratic Stability for Solving Stiff Differential Systems

This article proposes a new class of general linear method with $p=q$ and $r=s=p+1$. The construction of the present method is carried out using order conditions and error minimization subject to $A$- stability constraints. The proposed time integration schemes are $A$- and $L$-stable general linear methods (GLMs) equipped with inherent quadratic stability (IQS) criteria. We construct implicit GLMs of orders up to four with $p = q$ and $s = r$ along with the Nordsieck input vector assumption. Further, we test these schemes on three real-world problems: the van der Pol oscillator and two partial differential equations consisting of diffusion (Burgers' equation and the Gray-Scott model), and numerical results are presented. Computational results confirm that our proposed schemes are competitive with the existing GLMs and can be recognized as an alternative time integration scheme. We demonstrate the order of accuracy and convergence for the proposed schemes through observed order computation and error versus step size plots.