Adaptive mesh point selection for the efficient solution of scalar IVPs

TL;DR

This paper presents an adaptive mesh point selection method for scalar initial value problems that reduces local errors and improves efficiency, providing formulas for error reduction and demonstrating practical benefits through numerical examples.

Contribution

It introduces both nonconstructive and constructive adaptive mesh selection techniques that optimize local error reduction in scalar IVPs.

Findings

Significant error reduction factor achieved with adaptive meshes

Formulas quantify gain in terms of subintervals and error levels

Numerical example demonstrates practical effectiveness

Abstract

We discuss adaptive mesh point selection for the solution of scalar IVPs. We consider a method that is optimal in the sense of the speed of convergence, and aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.