The thesis numerically investigates the stability and convergence properties of the Parareal algorithm when it is run on the unsteady Stokes equations. The Parareal algorithm offers a parallel-in-time scheme for solving time dependent differential equations. The strategy of the algorithm follows the principles of domain decomposition.
The unsteady (or time dependent) Stokes equations are a set of partial differential equations (PDEs) that describe creeping flow. They are an important simplification of the more complex Navier-Stokes equations central to fluid dynamics. The motivation for wanting to use Parareal with the unsteady Stokes equations is to obtain faster computations in the temporal domain.
The stability and convergence are tested by using variations of the theta-rule for discretizing the temporal domain. The results were compared to similar numerical tests of the algorithm used with the heat equation. Prior analyses show that the algorithm will display proper convergence and stability traits for this equation. For stiff systems of ODEs the Parareal analysis states that the algorithm is stable when the coarse propagator uses theta in the range [2/3,1]. The unsteady Stokes equations are parabolic PDEs, and when semi-discretized in space they become systems of stiff ODEs. We therefore believe that the Parareal algorithm will remain stable and convergent when run on this problem.
Our numerical results indicate that the Parareal algorithm is indeed stable for [2/3,1] when it is used to solve the unsteady Stokes equations, although some uncertainty on its convergence rate is experienced at 2/3.
The thesis also numerically investigates the common estimate of the current error in the algorithm, which is used to determine convergence. We have performed numerical tests that indicate that the ratio of the true error and the approximative error is constant, which suggests that this is indeed a good estimate of the error at iteration k.
The thesis was solved using a combination of Python and the C++ library Diffpack, where all governing code is written in Python. Through its successful usage in this project, the thesis implementation acts as a proof-of-concept to that such a combination is indeed possible for solving problems like the unsteady Stokes equations.