On algebraic stability of general linear methods and peer methods

By definition algebraic stability of general linear methods is characterized by the existence of a weight matrix G leading to semi-definiteness of a 2×2 block test matrix depending on the coefficient matrices of the method. A congruence transformation is presented here reducing the number of places where G appears from 5 to 2 under assumptions satisfied by many methods from literature. A further reduction is possible to a test matrix depending on one single aggregated coefficient matrix P only. Simple sufficient and sharp necessary conditions on P are discussed. With these many algebraically stable implicit two-step peer methods with 3 stages and order 2 are constructed. Finally relations to Riccati equations and a generalized eigenvalue problem of Hill are discussed.