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

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.