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

TL;DR

The paper addresses high-order time integration for semi-explicit index-one DAEs by extending spectral deferred corrections (SDC) to a constrained variant (SDC-C) that integrates only the differential equations while enforcing algebraic constraints at every iteration. It establishes that, like ODE-SDC, SDC-C delivers a one-order-per-iteration improvement up to the underlying spectral quadrature order, and it demonstrates this both analytically (local truncation error) and numerically across linear, nonlinear, and PDAE test problems. The authors compare SDC-C with FI-SDC, SI-SDC, and classical Runge-Kutta solvers, showing competitive accuracy and substantial parallel efficiency when using diagonal preconditioners such as MIN-SR-NS; parallel SDC-C variants often outperform Radau IIA and DOPRI5 in runtime for comparable accuracy. The work provides practical evidence that constrained SDC offers a scalable, high-accuracy alternative for semi-explicit DAEs, with open-source implementation and data availability for reproducibility.

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.

Figures (13)

• Figure 1: Structure of induction proof. Green nodes indicate the cases shown by thm:lte_initial_SDC_C and present the base case for $k = 0$ and for all $m$. Orange nodes present cases proven by thm:lte_iters_first, they represent the base case for $m = 1$ and all iteration levels $k$. The red nodes are obtained finally by thm:lte_sdc_c.
• Figure 2: Spectral radius and eigenvalue distributions of iteration matrix of the constrained SDC scheme for the linear problem eq:linear_problem for $M = 6$ nodes. Left: Spectral radius for different time step sizes $\Delta t$. Right: Eigenvalue distributions (real part versus imaginary part) for $M = 6$ and $\Delta t = 0.01$.
• Figure 3: Local truncation error in $y$ and $z$ against time step sizes $\Delta t$ for the constrained SDC scheme using $\bm{Q}_\Delta^{\texttt{MIN-SR-NS}}$ for the linear problem eq:linear_problem for each iteration $k = 0,\dots,2M - 1$ with $M = 3$. Black dashed lines indicate the reference order. Left: Error in $y$. Right: Error in $z$.
• Figure 4: Wall-clock time against $L_{\infty}$ error for constrained SDC using $M = 6$ Radau IIA nodes for several choices of $\bm{Q}_\Delta$ across different time step sizes $\Delta t$.
• Figure 5: Wall-clock time versus $L_{\infty}$ error for different SDC variants using $M = 6$ collocation nodes for LU and MIN-SR-NS preconditioning compared with different Runge-Kutta methods for different time step sizes $\Delta t$.

Theorems & Definitions (19)

• remark thmcounterremark: Algebraic constraints as stiff limit
• remark thmcounterremark: Convergence in algebraic constraints
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof