Nordsieck Methods With Inherent Quadratic Stability

Abstract

We derive sufficient conditions which guarantee that the stability polynomial of Nordsieck method for ordinary differential equations has only two nonzero roots. Examples of such methods up to order four are presented which are A-and L-stable. These examples were obtained by computer search using the Schur criterion applied to the quadratic factor of the resulting stability polynomials.

Construction of Efficient General Linear Methods for Non-Stiff Differential Systems

“…In such a way, the resulting methods have larger stability regions compared to IRKS methods. A-and L-stable Nordsieck methods with inherent quadratic stability of the type (1.5) were investigated in [3]. The aim of our research is now to complete the investigation started in [3,10,11], by constructing explicit GLM of Nordsieck type with r = s = p, q = s − 1 and with p = q = s, r = s + 1, with quadratic stability and maximum area of the region of absolute stability.…”

“…In order to improve these results, the two-step Runge-Kutta methods with quadratic stability (QS), i.e. methods whose stability function has only two non-zero roots were introduced in [13,15] and successively developed in [3,10,11]. In [10,11] explicit Nordsieck methods of type (1.2) were proposed with p = q = s − 1, r = s with QS, i.e.…”

“…In recent years, classes of GLMs with optimal properties of efficiency and stability have been constructed and analyzed in the context of DIMSIMs (see for example [6] and also [5,15]), in the context of two-step Runge-Kutta methods [1,2,12,13,14,15] and in the context of GLMs in Nordsieck form [3,4,8,9,10,11,16,17]. A successful strategy for constructing high order and stable methods consists of imposing the order and the stage order conditions and then requiring that the stability function assumes a particular expression with desired stability properties.…”

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

“…To compare, we also present the results of numerical experiments of the L-stable Nordsieck GLM of order p = 4 given in [6] with the abscissa vector c = [ T . In Table 3 and Table 4, we have reported ns as the number of steps, nrs as the number of rejected steps, nJe as the number of Jacobian evaluations and ge as the global error at the end of the interval of integration for different given tolerances, tol.…”

“…In this paper, we are going to relax the concept of SIRKS to the concept of inherent quadratic stability (IQS). This property was first introduced in [19] for two-step Runge-Kutta (TSRK) methods and then presented for GLMs [6,14] which guarantees the stability function has only two nonzero roots. Using this approach, we solve fewer equations in comparison with methods based on SIRKS, which makes construction to be easier and gains some additional free parameters.…”