A note on the stability of rational Runge-Kutta methods
type: Item
instance of: scholarly article
title: A note on the stability of rational Runge-Kutta methods
zbMATH Open document ID: 0682.65034
DOI: 10.1007/BF02243228
author: Yanyan Li
published in: Computing
publication date: 1989-01-01T00:00:00Z
review text: Rational Runge-Kutta methods are studied for solution of a system of ordinary differential equations \(U'=f(U(t)),\) \(U(0)=u_ 0,\) \(f: R^ n\to R^ n.\) This method is called \(A_ 0\)-bounded if and only if there exists a constant \(\tau >0\) such that for every dimension \(n\geq 1\) \(| \bar u_{k+1}-u_{k+1}| \leq \tau | \bar u_ k-u_ k|\) holds where \(\{\) \(\bar u_ k\}\) and \(\{u_ k\}\) denote any two numerical sequences generated by applying the method to the problem \(U'=AU(t),\quad U(0)=u_ 0,\quad A=diag(\lambda_ 1,...,\lambda_ n)\) with the same (arbitrary choosen) step \(h>0,\quad \lambda_ i\leq 0\) and with different (arbitrary choosen) \(\bar u_ 0\), \(u_ 0\in R^ n.\) It is proved that there exists no rational Runge-Kutta method which is \(A_ 0\)-bounded.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4117759
zbMATH Keywords: stability
zbMATH Keywords: Rational Runge-Kutta methods
zbMATH Keywords: system
zbMATH Keywords: \(A_ 0\)-bounded
reviewed by: Miroslav Bartušek
MaRDI profile type: Publication
cites work: Q3746811
cites work: On the implementation of implicit Runge-Kutta methods
cites work: Q4725642
cites work: Q3687624
cites work: Unconditionally stable explicit methods for parabolic equations
cites work: Q4092887
cites work: Q5652137
cites work: Rational Runge-Kutta methods are not suitable for stiff systems of ODEs
cites work: A note on contractivity in the numerical solution of initial value problems
cites work: Rational Runge-Kutta methods for solving systems of ordinary differential equations
full work available at URL: https://doi.org/10.1007/bf02243228
OpenAlex ID: W159743449
description: scientific article; zbMATH DE number 4117759