Convergence results for the MATLAB ode23 routine
type: Item
instance of: scholarly article
title: Convergence results for the MATLAB ode23 routine
zbMATH Open document ID: 0920.65042
DOI: 10.1007/BF02510413
author: A. M. Stuart
author: Harbir Lamba
published in: BIT
publication date: 1999-04-27T00:00:00Z
review text: As it is well known most Runge-Kutta routines actually used in the numerical solution of ordinary differential equations (ODEs) allow the stepsize to change during the integration while standard existing theories assume a fixed stepsize along the integration. As a consequence of this, the analysis of variable stepsize methods under realistic stepsize changing techniques has been the subject of many papers in the last years. The authors study the convergence properties of the ode23 routine supplied in MATLAB Version 4.2. This routine is based on the embedded Fehlberg pair of Runge-Kutta formulas of orders 2 and 3 operating in extrapolation mode, i.e. the third order is used as advancing formula and the local error bound estimated by the difference between second and third order formulas is used to monitor the stepsize. Here the convergence is controlled by a user defined tolerance \(( \tau)\) and convergence means that the global error is bounded in the form \( C(B,T) \tau^{\gamma} \) for all sufficiently smooth initial value problems in \([0,T]\) with initial condition in a bounded set \(B\) and some \( \gamma >0\). In this setting the authors prove the convergence of the above variable stepsize ode23 routine provided that the solution holds away from points where the leading term of the local error expansion, scaled by \( \| f \| \), vanishes. More generally they show that this result also holds for all explicit embedded Runge-Kutta pairs \((p-1)p\) with \(p\) stages with the same timestep selection mechanism. In addition it is shown that the new ode23 routine supplied in MATLAB Version 5.0 which uses a new embedded Runge-Kutta pair and a more sophisticated time step selection algorithm does not fit into the above theory and further they present a two-dimensional linear system for which the new routine possesses very poor convergence properties.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 1251190
zbMATH Keywords: Runge-Kutta methods
zbMATH Keywords: variable stepsize methods
zbMATH Keywords: ode23 routine
zbMATH Keywords: MATLAB Version 4.2
zbMATH Keywords: embedded Fehlberg pair
zbMATH Keywords: error bound
zbMATH Keywords: convergence
zbMATH Keywords: MATLAB Version 5.0
reviewed by: Manuel Calvo
describes a project that uses: Matlab
describes a project that uses: Ode15s
describes a project that uses: ode113
describes a project that uses: MATLAB ODE suite
describes a project that uses: ode23
describes a project that uses: ode45
describes a project that uses: ode23s
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1251190