Special boundedness properties in numerical initial value problems
type: Item
instance of: scholarly article
title: Special boundedness properties in numerical initial value problems
zbMATH Open document ID: 1239.65045
author: A. Mozartova
author: Willem Hundsdorfer
author: Marc N. Spijker
published in: BIT
publication date: 2012-01-10T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/16648
review text: This paper is concerned with the analysis of some nonlinear stability properties of discretization methods for the numerical solution of initial value problems (IVPs) for differential systems. The proposed theory allows the authors to deal with standard classes of discretization methods for IVPs such as linear multistep and Runge-Kutta methods in connection with many stability properties studied along the last three decades for different classes of problems such as total-variation diminishing, strong stability preservation and boundedness. The authors consider general IVPs in a vector space \(\mathbf{V}\) with the form (1) \(\dot{u} (t) = F(t, u(t)),\) \( t \geq 0\), \( u(0)= u_0 \in \mathbf{V}\), and discretization methods that can be written as \((2) y_i = \sum_{j=1}^l \sigma_{ij} x_j + \Delta t \; \sum_{j=1}^m \tau_{ij} F( \overline{t}_j, y_j)\), \( i=1, \dots ,m\), where \( x_j \in \mathbf{V}\), \(j=1, \dots ,l\) are the input vectors with the information on the solution in the current step, \( \Delta t\) the step size, \( y_j \simeq u ( \overline{t}_j)\), \(j=1, \dots ,m\) are approximations to the solution at the time levels \( \overline{t}_j\) and \( S = ( \sigma_{ij})\), \( T = ( \tau_{ij})\) real matrices that define the method. As remarked most standard methods can be written in the form (2) with suitable \(S\) and \(T\) matrices. In this setting, assuming that for given \( \tau_0 >0\), the inequality \( \| v + \tau_0 \; F( \overline{t}_j, v) \| \leq \| v \| \) holds for \( j=1, \dots, m\) and and all \( v \in \mathbf{V}\) the authors derive conditions on the step size guaranteeing bounds on the solution of (2) of type \( \| y_i \| \leq \mu_i \max_{j=1, \dots ,l} \| x_j \|\), \((i=1, \dots ,m)\) and \( \| y_i \| \leq \sum_{j=1}^l \mu_{ij} \| x_j \|\), \((i=1, \dots ,m)\) for non negative coefficients \( \mu_i\) and \( \mu_{ij}\) where the functional \( \| \cdot \| \) can be a seminorm or even more general sublinear functionals. Among the most relevant conc…
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L50
zbMATH DE Number: 5996280
zbMATH Keywords: initial value problems
zbMATH Keywords: method of lines
zbMATH Keywords: strong stability preserving
zbMATH Keywords: total-variation-bounded
zbMATH Keywords: monotonicity
zbMATH Keywords: linear multistep
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: total-variation diminishing
describes a project that uses: RODAS
MaRDI profile type: Publication
OpenAlex ID: W2130056581
cites work: On the Convergence of Numerical Solutions to Ordinary Differential Equations
cites work: Q4725642
cites work: Q4435815
cites work: Stepsize Restrictions for the Total-Variation-Diminishing Property in General Runge--Kutta Methods
cites work: High order strong stability preserving time discretizations
cites work: Strong stability-preserving high-order time discretization methods
cites work: Q4714145
cites work: Q3779682
cites work: On strong stability preserving time discretization methods
cites work: Representations of Runge--Kutta Methods and Strong Stability Preserving Methods
cites work: Strong stability preserving hybrid methods
cites work: On monotonicity and boundedness properties of linear multistep methods
cites work: Monotonicity-Preserving Linear Multistep Methods
cites work: Q4421310
cites work: Stepsize Conditions for Boundedness in Numerical Initial Value Problems
cites work: Finite Volume Methods for Hyperbolic Problems
cites work: High-order linear multistep methods with general monotonicity and boundedness properties
cites work: Efficient implementation of essentially nonoscillatory shock-capturing schemes
cites work: Contractivity in the numerical solution of initial value problems
cites work: Stepsize Conditions for General Monotonicity in Numerical Initial Value Problems
DOI: 10.1007/S10543-011-0349-X
description: scientific article; zbMATH DE number 5996280