Towards a reliable implementation of least-squares collocation for higher-index differential-algebraic equations

TL;DR

This paper addresses the challenges of implementing least-squares collocation methods for higher-index differential-algebraic equations, proposing robust techniques and error estimates to improve numerical stability and reliability.

Contribution

It introduces a robust basis and collocation point selection, along with new error estimates, to enhance the implementation of collocation methods for higher-index DAEs.

Findings

Robust basis functions improve stability

New error estimates support method design

Implementation procedures enhance reliability

Abstract

In this note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher-index differential algebraic equations (DAEs). Since higher-index DAEs lead to ill-posed problems in natural settings, the dicrete counterparts are expected to be very sensitive, what attaches particular importance to their implementation. We provide a robust selection of basis functions and collocation points to design the discrete problem and substantiate a procedure for its numerical solution. Additionally, a number of new error estimates are proven that support some of the design decisions.