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 F-property. The proposed methods are characterized by \(r=s=p+1\) and satisfy inherent quadratic stability criteria. GLMs with the F-property offer a natural extension of Runge–Kutta schemes with the 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 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 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.
