Construction of Nordsieck Second Derivative General Linear Methods with Inherent Quadratic Stability

This paper describes the construction of second derivative general linear methods in Nordsieck form with stability properties determined by quadratic stability functions. This is achieved by imposing the so–called inherent quadratic stability conditions. After satisfying order and inherent quadratic stability conditions, the remaining free parameters are used to find the methods with L–stable property. Examples of methods with p = q = s = r − 1 up to order four are given

STABILITY ANALYSIS OF GENERAL LINEAR NYSTROM METHODS

In this talk we investigate the linear stability properties of the new family of General Linear Nystrom methods (GLNs), which is an extension of General Linear Methods to special second order ODEs y'' = f (x, y). We present the extension of the classical notions of stability matrix, stability polynomial, stability and periodicity interval, A-stability and P-stability to the family of GLNs. We next focus our interest on the derivation of highly stable GLNs inheriting the same stability properties of highly stable numerical methods existing in literature, i.e. Runge-Kutta-Nystrom methods based on indirect collocation on Gauss-Legendre points: this property, in analogy to a similar feature introduced for General Linear Methods solving first order ODEs, is called Runge-Kutta-Nystrom stability. The stability properties of GLNs with Runge-Kutta-Nystrom stability depend on a quadratic polynomial, which is exactly the stability polynomial of the best Runge-Kutta-Nystrom assumed as reference. We also provide examples of GLNs with Runge-Kutta-Nystrom stability