Spectral deferred corrections parallelized across the method for differential-algebraic equations
Abstract
In this work, the performance of different spectral deferred corrections (SDC) methods applied to initial value problems for differential-algebraic equations (DAEs) of index one is analyzed. The SDC method solves a series of correction equations, and after each iteration, the numerical solution is corrected by adding the obtained approximation of the error. The formulation of the SDC method allows for a parallelization across the method to obtain small-scale parallelism, for which a number of processes equal to the number of collocation nodes can be used. Recently, an analytical approach to finding optimal diagonal coefficients for parallel SDC was proposed. So far, no analysis with the new coefficients for parallel SDC applied to DAEs was done. We demonstrate that parallel SDC methods solve DAE problems with high accuracy faster than the associated sequential SDC schemes, obtaining speedup in a small-scale parallel regime.
Related
- Boundary value problems for linear differential-algebraic equations: solvability via the Kronecker canonical form
- Mathematical modeling, simulation and optimization via (port-Hamiltonian) differential-algebraic equations
- Strong convergence rates of stochastic theta Milstein methods for index-1 stochastic differential algebraic equations under non-globally Lipschitz conditions
- New bounds for the support of input-output equations in differential-algebraic systems
- Functional Analysis and Partial Differential Equations in Spectral Barron Spaces
BibTeX
@misc{bolten2026spectral,
title = {{Spectral deferred corrections parallelized across the method for differential-algebraic equations}},
author = {Matthias Bolten and Lisa Wimmer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.25908},
eprint = {2606.25908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25908},
}
RIS
TY - UNPB
TI - Spectral deferred corrections parallelized across the method for differential-algebraic equations
AU - Matthias Bolten
AU - Lisa Wimmer
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2606.25908
UR - https://arxiv.org/abs/2606.25908
AB - In this work, the performance of different spectral deferred corrections (SDC) methods applied to initial value problems for differential-algebraic equations (DAEs) of index one is analyzed. The SDC method solves a series of correction equations, and after each iteration, the numerical solution is corrected by adding the obtained approximation of the error. The formulation of the SDC method allows for a parallelization across the method to obtain small-scale parallelism, for which a number of processes equal to the number of collocation nodes can be used. Recently, an analytical approach to finding optimal diagonal coefficients for parallel SDC was proposed. So far, no analysis with the new coefficients for parallel SDC applied to DAEs was done. We demonstrate that parallel SDC methods solve DAE problems with high accuracy faster than the associated sequential SDC schemes, obtaining speedup in a small-scale parallel regime.
ER -