The aim of our research is the construction and analysis of efficient general linear methods (GLM), which achieve a good balance between accuracy and stability properties. In order to reach our goal we consider the class of GLMs with quadratic stability (QS), i.e. methods whose stability function has only two non-zero roots [4]. This property simplifies the study of stability and the search for methods with high order and good stability properties. In this talk we describe the conditions which guarantee the QS property and the construction of explicit Nordsieck methods with QS and maximum area of the region of absolute stability [2]. The search for these methods with high order is realized by various optimization routines [3], and the analogous search for another class of GLMs has been carried out in [1]. Examples of methods which compare favorably with respect to existing explicit GLM are presented, up to order six. Some issues concerning the implementation of our methods in a variable-step algorithm are addressed, such as the estimate of the local error and the computation of the input vector for the next step. This is a joint work with G. Izzo from Università di Napoli 'Federico II' and Z. Jackiewicz from Arizona State University. [1] M. Bras, A. Cardone, Construction of Efficient General Linear Methods for Non-Stiff Differential Systems, Math. Model. Anal. 17, 171-189 (2012). [2] A. Cardone, Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (1), 1{25 (2012). [3] A. Cardone, Z. Jackiewicz, H. D. Mittelmann, Optimization-based search for Nordsieck methods of high order with quadratic stability, to appear in Math. Model. Anal. [4] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley, Hoboken, New Jersey, 2009.
Efficient general linear methods for non-stiff differential equations
CARDONE, Angelamaria
2012-01-01
Abstract
The aim of our research is the construction and analysis of efficient general linear methods (GLM), which achieve a good balance between accuracy and stability properties. In order to reach our goal we consider the class of GLMs with quadratic stability (QS), i.e. methods whose stability function has only two non-zero roots [4]. This property simplifies the study of stability and the search for methods with high order and good stability properties. In this talk we describe the conditions which guarantee the QS property and the construction of explicit Nordsieck methods with QS and maximum area of the region of absolute stability [2]. The search for these methods with high order is realized by various optimization routines [3], and the analogous search for another class of GLMs has been carried out in [1]. Examples of methods which compare favorably with respect to existing explicit GLM are presented, up to order six. Some issues concerning the implementation of our methods in a variable-step algorithm are addressed, such as the estimate of the local error and the computation of the input vector for the next step. This is a joint work with G. Izzo from Università di Napoli 'Federico II' and Z. Jackiewicz from Arizona State University. [1] M. Bras, A. Cardone, Construction of Efficient General Linear Methods for Non-Stiff Differential Systems, Math. Model. Anal. 17, 171-189 (2012). [2] A. Cardone, Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (1), 1{25 (2012). [3] A. Cardone, Z. Jackiewicz, H. D. Mittelmann, Optimization-based search for Nordsieck methods of high order with quadratic stability, to appear in Math. Model. Anal. [4] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley, Hoboken, New Jersey, 2009.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
