An extension of general linear methods

Abstract

General Linear Methods (GLMs) were introduced as the natural generalizations of the classical Runge-Kutta and linear multistep methods. An extension of GLMs, so-called SGLMs (GLM with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. In this paper, we introduce the definitions of consistency, stability and convergence for an SGLM. It will be shown that in SGLMs, stability and consistency together are equivalent to convergence. Also, by introducing a subclass of SGLMs, we construct methods of this subclass up to the maximal order which possess Runge-Kutta stability property and A-stability for implicit ones.

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

“…Although GLMs include linear multistep methods, Runge-Kutta and many other standard methods, but they don't cover second derivative methods. So, GLMs were extended to second derivative general linear methods (SGLMs) by Butcher and Hojjati in [10] and studied more by Abdi and Hojjati [2,3,4].…”

“…Construction and the main features of SGLMs including pre-consistency, consistency, stability and types of these methods have been discussed in [3,10]. To construct efficient SGLMs for stiff ODEs with lower implementation cost, we will always assume that the coefficient matrices A and A have the form…”

“…Also, to ensure that the method (1.1) is zero-stable [3], here we will assume that the coefficient matrix V has the form…”

“…If the stability function has only one nonzero root, then the method is said to possess Runge-Kutta stability (RKS) property. In [3] Abdi and Hojjati introduced a subclass of SGLMs as SDIMSIMs (second derivative diagonally implicit multistage integration methods) and constructed methods of this subclass with RKS property. They obtained order barriers for some types of SGLMs which have RKS property in [3,4] and constructed SDIMSIMs of all types up to order 4 in [2].…”

“…In [3] Abdi and Hojjati introduced a subclass of SGLMs as SDIMSIMs (second derivative diagonally implicit multistage integration methods) and constructed methods of this subclass with RKS property. They obtained order barriers for some types of SGLMs which have RKS property in [3,4] and constructed SDIMSIMs of all types up to order 4 in [2]. Also, the construction of Nordsieck SGLMs up to order four and their implementation in a variable stepsize environment have been investigated in [5].…”