Convergence results for the MATLAB ode23 routine
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.
scientific article; zbMATH DE number 1251190
scientific article