The work to be presented in this talk focuses on a time-simultaneous multigrid algorithm for
the one-dimensional convection-diffusion equation, especially in the regime of small diffusion
coefficients. For spatial discretization we use a finite difference discretization, while the time
integrator is given, e.g., by the Crank-Nicolson scheme. By blocking all time steps into a global
linear system of equations and rearranging the degrees of freedom, we obtain a space-only
problem with vector-valued unknowns for each spatial node. Then standard methods, like the
Block Jacobi method or the generalized minimal residual method (GMRES) with preconditioning, can be used for the numerical solution of the (spatial) problem and allows parallelization in
space. We consider a time-simultaneous multigrid algorithm, which uses intergrid transfer operators and the solution techniques mentioned above for smoothing purposes. By treating more
time steps simultaneously, the dimension of the system of equations significantly increases
leading to a larger number of degrees of freedom per spatial unknown. This can be exploited to
achieve a higher degree of parallelism and the efficient use of many processors. In numerical
studies, the iterative multigrid solution of a problem with up to thousands of time steps is analyzed. For the heat equation, it can be observed that the number of iterations is independent of
the number of blocked time steps, the time step size, and the spatial resolution. Unfortunately,
stability issues arise if the diffusion coefficient is small compared to the grid size. Therefore,
new stabilization techniques are discussed, which remove artificial oscillations in the solution
and intend to improve the convergence behavior of the iterative solution algorithm.