Spectral deferred correction methods for second-order problems

TL;DR

Spectral Deferred Corrections (SDC) are iterative, collocation-based methods for solving ODEs. This work extends SDC to second-order initial value problems via a velocity-Verlet base method, establishing the first rigorous convergence and stability theory for second-order SDC and validating it with a Penning-trap Lorentz system. The central results show that each SDC iteration increases the global order by one if the force depends on velocity, or by two if it does not, with explicit local- and global-error bounds and stability domains analyzed through a damped oscillator model. Numerical experiments confirm the predicted convergence rates and demonstrate that SDC can outperform Picard iterations and, with enough iterations, rival RKN-4 in work-precision, while also exhibiting favorable energy-conservation properties due to its collocation structure. The approach is implemented in the public pySDC framework, highlighting practical applicability for high-accuracy simulations of second-order dynamics.

Abstract

Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nyström method.

Figures (7)

• Figure 1: Stability domain for $K=50$ iterations (upper left) and convergence domain (upper right) of SDC with $M=3$ Gauss-Legendre quadrature nodes. Stability domain of Picard iteration with $K=50$ iterations and $M=3$ nodes (lower left) and stability domain of RKN-4 (lower right).
• Figure 1: Absolute local error $\Delta x_{1}^{(\mathrm{abs})}$ in the first component of the particle's position (left) and velocity (right) using $K=1,2,3$ SDC iterations and M=5.
• Figure 1: Relative position error in $x_{1}$-direction (left) and $x_{3}$-direction (right) for SDC (solid lines), Picard (dashed lines) with different iteration numbers $K$ with $M=5$ quadrature nodes and RKN-4 against the total number of $f$ evaluations.
• Figure 2: Stability domains of SDC with $M=3$ Gauss-Legendre nodes and $K=1,2,3,4$ iterations.
• Figure 2: Absolute local error $\Delta x_{3}^{(\mathrm{abs})}$ in the third component of the particle's position (left) and velocity (right) using one, two and three SDC iterations and 5 quadrature nodes. In line with Theorem \ref{['the:main']}, the order increases by two per iteration.

Theorems & Definitions (15)

• Proposition 2.1
• Proof 1
• Remark 2.2
• Remark 2.3
• Proposition 3.1
• Proof 2
• Theorem 4.1
• Proof 3
• Theorem 4.2
• Proposition A.1