English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Two of the most popular parallel-in-time methods are Parareal andmultigrid-reduction-in-time (MGRIT). Recently, a general convergence theory wasdeveloped in Southworth (2019) for linear two-level MGRIT/Parareal thatprovides necessary and sufficient conditions for convergence, with tight boundson worst-case convergence factors. This paper starts by providing a new andsimplified analysis of linear error and residual propagation of Parareal,wherein the norm of error or residual propagation is given by one over theminimum singular value of a certain block bidiagonal operator. New discussionis then provided on the resulting necessary and sufficient conditions forconvergence that arise by appealing to block Toeplitz theory as in Southworth(2019). Practical applications of the theory are discussed, and the convergencebounds demonstrated to predict convergence in practice to high accuracy on twostandard linear hyperbolic PDEs: the advection(-diffusion) equation, and thewave equation in first-order form.

Tight two-level convergence of Linear Parareal and MGRIT: Extensions and implications in practice