On the analysis of spectral deferred corrections for differential-algebraic equations of index one

TL;DR

This paper introduces a new spectral deferred correction (SDC) scheme for semi-explicit differential-algebraic equations (DAEs) of index one, emphasizing parallelization and high accuracy, and compares it favorably with existing methods.

Contribution

A novel SDC scheme for semi-explicit DAEs that enforces algebraic constraints iteratively and supports parallel computation, extending SDC efficiency to DAEs.

Findings

The new SDC scheme achieves one order of convergence per iteration.

It is competitive with Runge-Kutta methods in accuracy.

Parallelized versions are highly efficient compared to other SDC methods.

Abstract

In this paper, we present a new SDC scheme for solving semi-explicit DAEs with the ability to be parallelized in which only the differential equations are numerically integrated is presented. In Shu et al. (2007) it was shown that SDC for ODEs achieves one order per iteration. We show that this carries over to the new SDC scheme. The method is derived from the approach of spectral deferred corrections and the idea of enforcing the algebraic constraints without numerical integration as in the approach of $\varepsilon$-embedding in Hairer and Wanner (1996). It enforces the algebraic constraints to be satisfied in each iteration and allows an efficient solve of semi-explicit DAEs with high-accuracy. The proposed scheme is compared with other DAE methods. We demonstrate that the proposed SDC scheme is competitive with Runge-Kutta methods for DAEs in terms of accuracy and its parallelized versions are very efficient in comparison to other SDC methods.