  Based on current trends in computer architectures, faster compute speeds must
come from increased parallelism rather than increased clock speeds, which are
currently stagnate. This situation has created the well-known bottleneck for
sequential time-integration, where each individual time-value (i.e., time-step)
is computed sequentially. One approach to alleviate this and achieve
parallelism in time is with multigrid. In this work, we consider
multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time
dimension that computes multiple time-steps in parallel. Like all multigrid
methods, MGRIT relies on the complementary relationship between relaxation on a
fine-grid and a correction from the coarse grid to solve the problem. All
current MGRIT implementations are based on unweighted-Jacobi relaxation; here
we introduce the concept of weighted relaxation to MGRIT. We derive new
convergence bounds for weighted relaxation, and use this analysis to guide the
selection of relaxation weights. Numerical results then demonstrate that
non-unitary relaxation weights consistently yield faster convergence rates and
lower iteration counts for MGRIT when compared with unweighted relaxation. In
most cases, weighted relaxation yields a 10%-20% saving in iterations. For
A-stable integration schemes, results also illustrate that under-relaxation can
restore convergence in some cases where unweighted relaxation is not
convergent.
