Search for Highly Stable General Linear Methods for Ordinary Differential Equations

In this talk we describe the construction of highly stable general linear methods (GLMs) for the numerical solution of ordinary differential equations (ODEs). We describe the construction of some classes of GLMs which are A-stable and L-stable using the Schur criterion, and algebraically stable methods using criteria proposed recently by Hill, Nonlinear stability of general linear methods, Numer. Math. 103(2006), 611–629, and Hewitt and Hill, Algebraically stable general linear methods and the G-matrix, to appear in BIT. We illustrate the results for the class of two-step Runge-Kutta methods with inherent Runge-Kutta stability for which one of the coefficient matrices is assumed to have a one-point spectrum. We also describe our search for algebraically stable methods in this class without imposing any restrictions on the coefficient matrices.