We describe the construction of general linear methods in Nordsieck form of order p and stage order q = p with large regions of absolute stability. We review the concepts of Runge-Kutta and quadratic stability and inherent Runge-Kutta and inherent quadratic stability [3] which aid in the construction of general linear methods with desirable stability properties. We also derive the representation formulas for some of the coefficient matrices of Nordsieck methods [1]. The search for these methods is based on maximizing the area of the intersection of the region of absolute stability with the negative complex plane by various optimization routines [2]. We first construct quadratic polynomials with large regions of absolute stability and then search for methods whose stability functions matches these stability polynomials. The efficient computation of coefficients of these polynomials utilizes the fast Fourier transform. Examples of methods obtained in this way are presented up to the order six. This is a joint work with A. Cardone from University of Salerno, and H. Mittelmann from Arizona State University. REFERENCES [1] A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Submitted. [2] A. Cardone, Z. Jackiewicz and H. Mittelmann. Search for Nordsieck methods of high order with quadratic stability. Manuscript. [3] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New Jersey, 2009.

General Linear Methods with Large Regions of Absolute Stability

CARDONE, Angelamaria;
2011

Abstract

We describe the construction of general linear methods in Nordsieck form of order p and stage order q = p with large regions of absolute stability. We review the concepts of Runge-Kutta and quadratic stability and inherent Runge-Kutta and inherent quadratic stability [3] which aid in the construction of general linear methods with desirable stability properties. We also derive the representation formulas for some of the coefficient matrices of Nordsieck methods [1]. The search for these methods is based on maximizing the area of the intersection of the region of absolute stability with the negative complex plane by various optimization routines [2]. We first construct quadratic polynomials with large regions of absolute stability and then search for methods whose stability functions matches these stability polynomials. The efficient computation of coefficients of these polynomials utilizes the fast Fourier transform. Examples of methods obtained in this way are presented up to the order six. This is a joint work with A. Cardone from University of Salerno, and H. Mittelmann from Arizona State University. REFERENCES [1] A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Submitted. [2] A. Cardone, Z. Jackiewicz and H. Mittelmann. Search for Nordsieck methods of high order with quadratic stability. Manuscript. [3] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New Jersey, 2009.
9789984453514
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4423253
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