On the Relation Between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential Equations

1956 ◽  
Vol 4 (1) ◽  
pp. 20-37 ◽  
Author(s):  
Jim Douglas, Jr.
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Stability And Convergence ◽  
Hyperbolic Differential Equations

Numerical Solution of the Coupled System of Nonlinear Fractional Ordinary Differential Equations

2017 ◽  
Vol 9 (3) ◽  
pp. 574-595 ◽  
Author(s):  
Xiaojun Zhou ◽  
Chuanju Xu
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Coupled System ◽  
High Order ◽  
Stability And Convergence ◽  
General Technique ◽  
Smoothness Assumption ◽  
Nonlinear Coupled System

AbstractIn this paper, we consider the numerical method that is proposed and analyzed in [J. Cao and C. Xu, J. Comput. Phys., 238 (2013), pp. 154–168] for the fractional ordinary differential equations. It is based on the so-called block-by-block approach, which is a common method for the integral equations. We extend the technique to solve the nonlinear system of fractional ordinary differential equations (FODEs) and present a general technique to construct high order schemes for the numerical solution of the nonlinear coupled system of fractional ordinary differential equations (FODEs). By using the present method, we are able to construct a high order schema for nonlinear system of FODEs of the orderα,α>0. The stability and convergence of the schema is rigorously established. Under the smoothness assumptionf,g∈C4[0,T], we prove that the numerical solution converges to the exact solution with order 3+αfor 0<α≤1 and order 4 forα>1. Some numerical examples are provided to confirm the theoretical claims.

scholarly journals A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 914
Author(s):  
Zarina Bibi Ibrahim ◽  
Amiratul Ashikin Nasarudin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Stability And Convergence ◽  
Stiff Differential Equations ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Backward Differentiation Formulas ◽  
The Stability ◽  
Differentiation Formulas

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

scholarly journals Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Haiyan Yuan ◽  
Jingjun Zhao ◽  
Yang Xu
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Nonlinear Stability ◽  
Stability And Convergence ◽  
Numerical Example ◽  
Runge Kutta ◽  
Lagrangian Interpolation ◽  
The Stability ◽  
Theoretical Results

This paper is devoted to the stability and convergence analysis of the Additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of multidelay-integro-differential equations (MDIDEs). GDN-stability and D-convergence are introduced and proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASI-stability give GDN-stability. A numerical example is given to illustrate the theoretical results.

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