The spatial discretization of time-dependent partial differential equations by finite elements, finite difference or finite volumes leads to systems of ordinary differential equations of very large dimension. Such systems can no longer be solved efficiently by classical ODE software. Their solution requires specialized solvers that take the structure of the problems into account. When using higher order implicit Runge-Kutta or Boundary Value Method time-discretization schemes, the size of the system to be solved in every time step amounts to a multiple of the number of spatial unknowns. We will show in this talk that these systems can be solved very efficiently, with a complexity that is linear in the number of unknowns when multigrid PDE-algorithms are used.

We will present in particular an algebraic multigrid algorithm fully coupled implicit discretizations of the time-dependent diffusion and curlcurl equations. The algorithm uses a blocksmoother, updating all stage values related to a grid point simultaneously. The multigrid hierarchy can be derived from the hierarchy built by any suitable AMG algorithm for the stationary version of the problem considered. By a theoretical analysis and numerical experiments, we show that the convergence of the algorithm is similar to the convergence of the stationary AMG algorithm on which it is based.