Effect of space discretization on the parareal algorithm for advection-diffusion equations

Abstract

The influence of time-integrator on the convergence rate of the parallel-in-time algorithm parareal has been extensively studied in literature, but the effect of space discretization was only rarely considered. In this paper, using the advection–diffusion equation parametrized by a diffusion coefficient $\nu >0$ as the model, we show that the space discretization indeed has a non-negligible effect on the convergence rate, especially when $\nu$ is small. In particular, for two space discretizations—the centered FD (finite difference) method and a Compact FD method of order 4, we show that the algorithm converges with very different rates, even though both the coarse and fine solvers of the algorithm are strongly stable under these two space discretizations. Numerical results for one-dimensional and two-dimensional cases are presented to validate the theoretical predictions.