Parallelizing spectral deferred corrections across the method
This work addresses the computational bottleneck of serial Gauss-Seidel preconditioners in SDC, offering parallel solutions for high-performance computing in time integration.
The paper presents two strategies to parallelize spectral deferred corrections (SDC) across collocation nodes, enabling simultaneous updates. For linear problems, parallel preconditioners achieve up to 90% parallel efficiency, while diagonalization with inexact Newton methods shows convergence rates independent of time-step size for nonlinear problems.
In this paper we present two strategies to enable "parallelization across the method" for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gauss-Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem.