General Linear Method with F-property and Inherent Quadratic Stability for Solving Stiff differential systems

This work introduces a new class of general linear methods (GLMs) for solving systems of time-dependent differential equations, based on the Nordsieck input vector and equipped with the \emph{F}-property. The proposed methods are characterized by $r=s=p+1$ and satisfy inherent quadratic stability criteria. GLMs with the \emph{F}-property offer a natural extension of Runge–Kutta schemes with the \emph{first same as last} (FSAL) property and provide improved efficiency over non-FSAL methods with the same number of stages. We develop implicit GLMs with the \emph{F}-property that are well suited for stiff differential systems arising from semi-discretization of partial differential equations (PDEs). The theoretical framework needed for constructing these schemes is presented, along with a key modification in the matrix equivalence necessary for enforcing IQS in the presence of the \emph{F}-property. The proposed classes are then tested on three different test problems. The results of the numerical simulations carried out for all the three problems reveal a good agreement with the reference solutions. The results are interpreted using computation of error norms, estimated orders, and work precision diagrams.