General linear methods are a wide class of numerical methods for solving ordinary differential systems which includes many classical methods, such as for example Runge-Kutta or linear multistep methods. We describe the construction of explicit methods in Nordsieck form with the so-called quadratic stability (QS), i.e. methods with stability function of the form p(w, z) = w^(r-1) (w^2 - p1(z)w + p0(z)); where p1(z) and p0(z) are polynomials of z. After satisfying order conditions and criteria that guarantee quadratic form of its stability function, the remaining free parameters of the methods are used to maximize region of absolute stability. We discuss the error estimation formulas and step changing strategies for constructed methods. The results of numerical experiments with variable stepsize code are also presented. REFERENCES [1] M. Bras. Nordsieck Methods with Inherent Quadratic Stability. Math. Model. Anal., 16 (1): 82-96. [2] M. Bras and A. Cardone. Construction of Efficient General Linear Methods for Non-Stiff Differential Systems. Math. Model. Anal., 17 (2): 171-189. [3] Z. Bartoszewski and Z. Jackiewicz. Explicit Nordsieck methods with extended stability regions. Appl. Math. Comput., 218 6056-6066. [4] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New Jersey, 2009.
Nordsieck methods for non-stiff ordinary differential equations
CARDONE, Angelamaria
2012-01-01
Abstract
General linear methods are a wide class of numerical methods for solving ordinary differential systems which includes many classical methods, such as for example Runge-Kutta or linear multistep methods. We describe the construction of explicit methods in Nordsieck form with the so-called quadratic stability (QS), i.e. methods with stability function of the form p(w, z) = w^(r-1) (w^2 - p1(z)w + p0(z)); where p1(z) and p0(z) are polynomials of z. After satisfying order conditions and criteria that guarantee quadratic form of its stability function, the remaining free parameters of the methods are used to maximize region of absolute stability. We discuss the error estimation formulas and step changing strategies for constructed methods. The results of numerical experiments with variable stepsize code are also presented. REFERENCES [1] M. Bras. Nordsieck Methods with Inherent Quadratic Stability. Math. Model. Anal., 16 (1): 82-96. [2] M. Bras and A. Cardone. Construction of Efficient General Linear Methods for Non-Stiff Differential Systems. Math. Model. Anal., 17 (2): 171-189. [3] Z. Bartoszewski and Z. Jackiewicz. Explicit Nordsieck methods with extended stability regions. Appl. Math. Comput., 218 6056-6066. [4] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New Jersey, 2009.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
