A parallel-in-time Newton's method-based ODE solver

TL;DR

This paper presents a new parallel-in-time Newton's method-based solver for nonlinear ODEs that leverages parallel prefix sums to achieve logarithmic span complexity, outperforming existing methods like Parareal.

Contribution

The paper introduces a novel parallel-in-time approach for nonlinear ODEs using Newton's method combined with parallel prefix sums, reducing computational complexity.

Findings

Achieves logarithmic span complexity in solving ODEs.

Demonstrates improved runtime over Parareal through numerical simulations.

Validates the method's efficiency on various ODE systems.

Abstract

In this article, we introduce a novel parallel-in-time solver for nonlinear ordinary differential equations (ODEs). We state the numerical solution of an ODE as a root-finding problem that we solve using Newton's method. The affine recursive operations arising in Newton's step are parallelized in time by using parallel prefix sums, that is, parallel scan operations, which leads to a logarithmic span complexity. This yields an improved runtime compared to the previously proposed Parareal method. We demonstrate the computational advantage through numerical simulations of various systems of ODEs.