Spectral deferred correction methods for second-order problems

Spectral deferred corrections (SDC) is an iterative method for the numerical solution of ordinary differential equations. It can be interpreted as a Picard iteration for the collocation problem, preconditioned with a low order method. SDC has been studied for first-order problems, using explicit, implicit, or implicit-explicit Euler as a preconditioner. It has been shown that SDC can achieve an arbitrarily high order of accuracy and possesses good stability properties.

In this talk, we will present an analysis of the convergence and stability properties of the SDC method when applied to second-order ODEs and using velocity-Verlet as a preconditioner. While a variant of this method called Boris-SDC for the Lorentz equation has been investigated, no general analysis of its properties for general second-order problems exists.

We will show that the order of convergence depends on whether the force on the right-hand side of the system depends on velocity (like in the Lorentz equation) or not (like in the undamped harmonic oscillator). Moreover, we also show that the SDC iteration is stable under certain conditions. We compare its stability domain with the Picard iteration and validate our theoretical analysis in numerical examples.

Keywords: Spectral deferred corrections(SDC), Picard iteration, collocation method, velocity-Verlet, preconditioner, stability, convergence.