scholarly journals CONSTRUCTION OF EFFICIENT GENERAL LINEAR METHODS FOR NON-STIFF DIFFERENTIAL SYSTEMS

2012 ◽  
Vol 17 (2) ◽  
pp. 171-189 ◽  
Author(s):  
Michal Bra´s ◽  
Angelamaria Cardone
Keyword(s):  
Absolute Stability ◽  
Stability Region ◽  
General Linear Methods ◽  
General Linear ◽  
Large Area ◽  
Differential Systems ◽  
Quadratic Stability ◽  
Free Parameters ◽  
The Stability ◽  
Region Of Absolute Stability

This paper describes the construction of explicit general linear methods in Nordsieck form with inherent quadratic stability and large areas of the stability region. After satisfying order and inherent quadratic stability conditions, the remaining free parameters are used to find the methods with large area of region of absolute stability. Examples of methods with p = q + 1 = s = r and p = q = s = r - 1 up to order 6 are given.

scholarly journals EXPLICIT GENERAL LINEAR METHODS WITH A LARGE STABILITY REGION FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

2019 ◽  
Vol 24 (4) ◽  
pp. 478-493
Author(s):  
Hassan Mahdi ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Stability Region ◽  
Quadrature Rule ◽  
General Linear Methods ◽  
General Linear ◽  
Large Area ◽  
Integral Term ◽  
Free Parameters ◽  
The Stability

In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods.

scholarly journals OPTIMIZATION-BASED SEARCH FOR NORDSIECK METHODS OF HIGH ORDER WITH QUADRATIC STABILITY POLYNOMIALS

2012 ◽  
Vol 17 (3) ◽  
pp. 293-308 ◽  
Author(s):  
Angelamaria Cardone ◽  
Zdzislaw Jackiewicz ◽  
Hans Mittelmann
Keyword(s):  
State Of The Art ◽  
High Order ◽  
General Linear Methods ◽  
General Linear ◽  
Stability Function ◽  
Optimization Software ◽  
Quadratic Stability ◽  
Nordsieck Methods ◽  
The Stability

We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7.

scholarly journals CONSTRUCTION OF NORDSIECK SECOND DERIVATIVE GENERAL LINEAR METHODS WITH INHERENT QUADRATIC STABILITY

2017 ◽  
Vol 22 (1) ◽  
pp. 60-77 ◽  
Author(s):  
Akram Movahedinejad ◽  
Gholamreza Hojjati ◽  
Ali Abdi
Keyword(s):  
General Linear Methods ◽  
General Linear ◽  
Stability Conditions ◽  
Second Derivative ◽  
Stable Property ◽  
Stability Properties ◽  
Quadratic Stability ◽  
Stability Functions ◽  
Free Parameters ◽  
Order Four

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.

scholarly journals EFFICIENT GENERAL LINEAR METHODS OF HIGH ORDER WITH INHERENT QUADRATIC STABILITY

2014 ◽  
Vol 19 (4) ◽  
pp. 450-468 ◽  
Author(s):  
Michal Bras ◽  
Zdzislaw Jackiewicz
Keyword(s):  
High Order ◽  
General Linear Methods ◽  
General Linear ◽  
Stability Conditions ◽  
Stability Function ◽  
Quadratic Stability ◽  
Stage Order

We search for general linear methods with s internal stages and r = s + 1 external stages of order p = s + 1 and stage order q = s. We require that stability function of these methods has only two non-zero roots. This is achieved by imposing the so-called inherent quadratic stability conditions. Examples of such general linear methods which are A- and L-stable up to the order p = 8 and stage order q = p - 1 are derived.

scholarly journals 3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
J. O. Ehigie ◽  
S. A. Okunuga ◽  
A. B. Sofoluwe
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Absolute Stability ◽  
Second Order ◽  
Stability And Convergence ◽  
Block Methods ◽  
Good Region ◽  
The Stability ◽  
The Individual ◽  
Region Of Absolute Stability

A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

scholarly journals ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

2019 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Hippolyte Séka ◽  
Kouassi Richard Assui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Absolute Stability ◽  
Stability Region ◽  
Kutta Method ◽  
Stability Regions ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Runge Kutta ◽  
The Stability

In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.

Highly Stable General Linear Methods for Differential Systems

2009 ◽  
Author(s):  
R. D’Ambrosio ◽  
G. Izzo ◽  
Z. Jackiewicz ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
General Linear Methods ◽  
General Linear ◽  
Differential Systems
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