Construction of algebraically stable DIMSIMs

The class of general linear methods for ordinary differential equations combines the advantages of linear multistep methods (high efficiency) and Runge–Kutta methods (good stability properties such as AA-, LL-, or algebraic stability), while at the same time avoiding the disadvantages of these methods (poor stability of linear multistep methods, high cost for Runge–Kutta methods). In this paper we describe the construction of algebraically stable general linear methods based on the criteria proposed recently by Hewitt and Hill. We also introduce the new concept of ϵϵ-algebraic stability and investigate its consequences. Examples of ϵϵ-algebraically stable methods are given up to order p=4p=4.