Parallelizing spectral deferred corrections across the method

TL;DR

This paper introduces two strategies to parallelize spectral deferred corrections (SDC), enabling simultaneous updates of collocation nodes, thus improving computational efficiency for solving differential equations.

Contribution

The paper presents novel parallelization strategies for SDC, including parallel preconditioners and diagonalization of the quadrature matrix, tested on multiple problems.

Findings

Parallel preconditioners improve convergence

Diagonalization enables simultaneous node updates

Effective for nonlinear diffusion problems

Abstract

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.