Algebraically stable two-step Runge-Kutta and continuous methods for ordinary differential equations

We investigate algebraic stability of two-step Runge-Kutta (TSRK) methods and of the new class of two-step almost collocation (TSAC) methods for ordinary differential equations. These continuous methods are obtained by modifying the classical collocation technique in order to achieve a better balance between strong stability properties and high uniform order of convergence on the whole interval of integration. We describe the search for algebraically stable methods using some criteria recently proposed by Hill [1,2]. In particular, by suitably adapting the idea of [1], an optimization-based numerical approach to derive the coefficients of algebraically stable TSAC methods is presented [3]. Because of the purely numerical nature of this approach, the coefficients of the corresponding methods are not expressed in rational form, but they are provided with a certain number of correct digits. As regards TSRK methods we are able to provide algebraically stable methods whose coefficients are expressed in rational form [4], by using the alternative approach proposed in [2]. This is a joint work with Z. Jackiewicz from Arizona State University, and B. Paternoster, R. D'Ambrosio from University of Salerno. [1] A.T. Hill, Nonlinear stability of general linear methods, Numer. Math. 103, 611{629 (2006). [2] L.L. Hewitt, A.T. Hill, Algebraically stable diagonally implicit general linear methods, Appl. Numer. Math. 60, 629-636 (2010). [3] D. Conte, R. D'Ambrosio, Z. Jackiewicz, B. Paternoster, Numerical search for algebraically stable two-step almost collocation methods, submitted. [4] D. Conte, R. D'Ambrosio, Z. Jackiewicz, B. Paternoster, A practical approach for the derivation of algebraically stable two-step Runge-Kutta methods, Math. Model. Anal. 17 (1), 65-77 (2012).