Convergence Analysis of the Parareal Algorithm with Nonuniform Fine Time Grid

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024.
Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.