Second derivative general linear methods with inherent Runge–Kutta stability

Explicit Nordsieck Second Derivative General Linear Methods for Odes

“…The sufficient conditions for Nordsieck SIRKS with are [34] for some matrix X , and Here, the relation means that the matrices and are identical except possibly in their first rows. The matrix X appearing in equation (2.1) takes the form Although in [34] the conditions for SIRKS have been obtained for , , it is easy to show that equation (2.1) together with equation (2.2) are sufficient conditions for arbitrary values of and .…”

“…, 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.…”

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

“…In [22], a special case of SGLMs as A-A-V methods was introduced which made easier to reach RKS property. Similar to what has been done in [8,11,12,13,33,34] for GLMs with inherent Runge-Kutta stability (IRKS), SGLMs with inherent Runge-Kutta stability (SIRKS) were introduced in [32] in which the coefficients matrices satisfy some relationships to ensure the methods have RKS property. In [1], SGLMs of high order with s = 2 were investigated which have quadratic stability (QS) property.…”