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

Abstract

Abstract. 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.Keywords: stiff differential equations, second derivative methods, Nordsieck methods, inherent quadratic stability, A-and L-stability.

Explicit Nordsieck Second Derivative General Linear Methods for Odes

“…, r − 3, respectively. However, the task is too complicated for the methods with large numbers of s and r. Nevertheless, some interrelationships among the coefficients matrices of the method have been obtained which are sufficient conditions to possess RKS [34] or QS [35]. These interrelationships are respectively referred to as IRKS or IQS conditions.…”

“…Due to the benefits of the class of second derivative methods [21, 22, 27, 28], to obtain methods with high order accuracy and good stability properties, GLMs were extended to second derivative general linear methods (SGLMs) by Butcher and Hojjati [17]. This large family of methods has been widely analysed and successfully implemented on various time-dependent problems [1–7, 34, 35]. These methods are characterized by p and q respectively as the order and stage order, r as the number of external stages, s as the number of internal stages, the abscissa vector and the coefficients matrices SGLMs, on the uniform grid , , with h as the stepsize and , take the following form: with g as the second derivative of the solution given by .…”

“…Movahedinejad et al [34] introduced SGLMs in Nordsieck form with IRKS. By relaxing the concept of IRKS to the concept of IQS, methods with such property in the classes of two-step Runge–Kutta (TSRK) methods [25], GLMs [13, 20] and SGLMs [35] have been introduced. SGLMs with IRKS and IQS will be respectively referred to as SIRKS and SIQS.…”