Detecting and locating a singular point in the numerical solution of IVPs for ODEs
Most of the available numerical methods for solving initial value problems for ordinary differential equations do not give good results in the neighborhood of a singularity. The objective is to recognize, at very little extra cost, when singularities are affecting the performance of a method. The authors develop a new approach which can be adopted by standard methods and which does not require special action or information from the user. The approach is comprised of two stages. The first one is a preliminary singularity detection stage, which is straightforward while the second stage confirms the existence of a singularity and attempts to approximate its location. There are three alternative techniques for the second stage. In the first two techniques, based on finding some rational approximation, the location of the singularity is approximated by a zero of the corresponding denominator polynomial, whereas the third technique approximates an interval containing the singularities. The approach can be used with any Runge-Kutta formula with an associated interpolant. The numerical results show that the techniques are effective in detecting and identifying the location of singularities. The approach is relatively inexpensive to implement and could be employed in standard initial value problems software to signal when a problem is better solved by special purpose methods designed to handle singular problems.
scientific article; zbMATH DE number 86177
scientific article