Efficient Nordsieck second derivative general linear methods: construction and implementation

Hojjati

2021

Preprint

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q = p, some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

“…In this subsection, we derive methods of order three with the abscissa vector c = [0 1 2 1] T . The coefficients matrices of these methods have the following form…”

Section: Methods Of Order P =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Variable stepsize SDIMSIMs for ordinary differential equations

Jalilian

Hojjati

2021

Preprint

“…Second derivative general linear methods (SGLMs) for the numerical solution of (1.1) as a general framework for the traditional second derivative methods are an extension of general linear methods (GLMs) [11,12,26]. These methods incorporating the second derivative of the solution into the formula have been studied, for instance, in [2,3,4,6,7,8,10,13]. SGLMs for solving (1.1) can be represented by abscissa vector…”

Section: Introductionmentioning

confidence: 99%

High Order Second Derivative Diagonally Implicit Multistage Integration Methods for Odes

Sharifi

Mathematical Modelling and Analysis

Braś

et al. 2023

Self Cite

Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.

“…where f : R m → R m , were introduced by Butcher (1966Butcher ( , 2016 and extended to second derivative general linear methods (SGLMs) to cover second derivative methods (see, for instance, Abdi 2016; Abdi and Behzad 2018;Abdi et al 2014;Abdi andHojjati 2011a, b, 2015). Izzo et al (2010) introduced GLMs for the numerical solution of VIEs (1).…”

Section: Introductionmentioning

confidence: 99%

General linear methods with large stability regions for Volterra integral equations

2019

Comp. Appl. Math.

Self Cite

General linear methods are well-known as a large family of methods for the numerical solution of Volterra integral equations of the second kind. This paper is concerned with the construction of such methods with a large region of order p and high stage order q = p with a large region of absolute stability. Some numerical results are presented which indicate the effectiveness of the proposed schemes.