Implicit-Explicit Integral Deferred Correction Methods for Stiff Problems

TL;DR

This paper analyzes the order reduction in implicit-explicit integral deferred correction methods for stiff singular perturbation problems, providing error estimates and numerical validation for Van der Pol and PDE cases.

Contribution

It offers a detailed error analysis revealing the order reduction phenomenon in InDC-IMEX methods applied to stiff problems with small parameters.

Findings

Error bounds for InDC-IMEX methods on SPPs

Order reduction phenomenon identified and explained

Numerical results confirm theoretical analysis

Abstract

The main goal of this paper is to investigate the order reduction phenomenon that appears in the integral deferred correction (InDC) methods based on implicit-explicit (IMEX) Runge-Kutta (R-K) schemes when applied to a class of stiff problems characterized by a small positive parameter $\varepsilon$, called singular perturbation problems (SPPs). In particular, an error analysis is presented for these implicit-explicit InDC (InDC-IMEX) methods when applied to SPPs. In our error estimate, we expand the global error in powers of $\varepsilon$ and show that its coefficients are global errors of the corresponding method applied to a sequence of differential algebraic systems. A study of these errors in the expansion yields error bounds and it reveals the phenomenon of order reduction. In our analysis we assume uniform quadrature nodes excluding the left-most point in the InDC method and the globally stiffly accurate property for the IMEX R-K scheme. Numerical results for the Van der Pol equation and PDE applications are presented to illustrate our theoretical findings.