The paper presents a numerical study of the convergence behavior of the time-parallel Parareal method for the heat equation with space- and time-dependent coefficients. It demonstrates that the good convergence of Parareal for diffusive problems is only marginally affected by both jumps in the diffusion coefficients and a diffusion coefficient that changes in time. For linear problems, Parareal can be interpreted as a preconditioned fixed point iteration and, at least for small enough problems, the iteration matrix and its maximum singular value can be computed numerically. An example is shown that demonstrates that the largest singular value gives a reasonable estimate for the convergence of Parareal. Extending the analysis presented here to more complicated cases e.g. in three dimensions with complicated geometries, with coefficient jumps not aligned with the mesh or cases that also include advection would be an interesting direction of future research.
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