TOWARD PARALLEL COARSE GRID CORRECTION FOR THE PARAREAL ALGORITHM.

In this paper, we present an idea toward parallel coarse grid correction (CGC) for the parareal algorithm. It is well known that such a CGC procedure is often the bottleneck of speedup of the parareal algorithm. For an ODE system with initial-value condition u(0) = u0 the idea can be explained as follows. First, we apply the G-propagator to the same ODE system but with a special condition u(0) = αu(T), where α ∈ ℝ is a crux parameter. Second, in each iteration of the parareal algorithm the CGC procedure will be carried out by the so-called diagonalization technique established recently. The parameter α controls both the roundoff error arising from such a diagonalization technique and the convergence rate of the resulting parareal algorithm. We show that there exists some threshold α* such that the parareal algorithm with diagonalization-based CGC possesses the same convergence rate as that of the parareal algorithm with classical CGC if |α| ≤ α*. With |α| = α*, we show that the condition number associated with the diagonalization technique is a moderate quantity of order O(1) (and therefore the roundoff error is small) and is independent of the length of the time interval. Numerical results are given to support our findings. [ABSTRACT FROM AUTHOR]