Stepsize restrictions for total-variation-boundedness in general Runge--Kutta procedures
type: Item
instance of: scholarly article
title: Stepsize restrictions for total-variation-boundedness in general Runge--Kutta procedures
zbMATH Open document ID: 1073.65076
author: L. Ferracina
author: Marc N. Spijker
published in: Applied Numerical Mathematics
publication date: 2005-04-21T00:00:00Z
review text: The Cauchy problem for the equation \[ {\partial u(x,t)\over\partial t}+ {\partial f(u(x,t))\over\partial x}= 0,\quad t\geq 0,\;-\infty< x<\infty, \] using the method of lines, can be regarded as an ordinary differential equation in the form \[ {dU(t)\over dt}= F(U(t)),\quad U(t_0)= u_0\tag{1} \] with \(U\in\mathbb{R}^\infty\). The authors apply the Runge-Kutta method to (1) and indicate a stepsize restriction \(\Delta t\leq\Delta t_0\), which guarantee total variation bounded or total variation diminishing for the approximations \(U_n\), obtained through this method. They use for the proof of the main theorem two coefficients \(R(A,b)\) and \(S(A,b)\) defined for an irreducible Runge-Kutta method. The practical calculation of these coefficients is difficult.
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 2160263
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Runge-Kutta method
zbMATH Keywords: total variation diminishing
zbMATH Keywords: total variation bounded
zbMATH Keywords: initial value problem
zbMATH Keywords: conservation law
zbMATH Keywords: method of lines
zbMATH Keywords: stepsize restriction
reviewed by: Ivan Secrieru
describes a project that uses: HE-E1GODF
describes a project that uses: RODAS
MaRDI profile type: Publication
full work available at URL: j.apnum.2004.08.024
OpenAlex ID: W2136610508
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DOI: 10.1016/J.APNUM.2004.08.024
description: scientific article; zbMATH DE number 2160263