SI-method for solving stiff nonlinear boundary value problems

TL;DR

This paper provides a comprehensive theoretical analysis of the SI-method for stiff boundary value problems, establishing conditions for its applicability, error estimates, and demonstrating its effectiveness through numerical examples and open-source implementation.

Contribution

The paper extends the SI-method framework, offering new applicability conditions, error analysis, and practical insights for solving stiff nonlinear boundary value problems.

Findings

The SI-method is stable and efficient for certain stiff boundary value problems.

Sufficient conditions for the method's applicability are established.

An open-source C++ implementation is provided for practical use.

Abstract

The paper contains a thorough theoretical analysis of the SI-method, which was firstly introduced in arXiv:1601.04272v8 and proved to be remarkably stable and efficient when applied to some instances of stiff boundary value problems (like the Troesch's problem). By suggesting a more general view on the SI-method's idea and framework, we managed to obtain sufficient conditions for the method to be applicable to a certain class of two-point boundary value problems. The corresponding error estimates are provided. Special attention is devoted to the exploration of the method's capabilities via a set of numerical examples. The implementation details of the method are discussed in fair depth. An open-source C++ implementation of the SI-method is freely available at the public repository https://github.com/imathsoft/MathSoftDevelopment.