Communications in Mathematical Sciences
Volume 16 (2018)
Overlapping localized exponential time differencing methods for diffusion problems
Pages: 1531 – 1555
The localized exponential time differencing (ETD) based on overlapping domain decomposition has been recently introduced for extreme-scale phase field simulations of coarsening dynamics, which displays excellent parallel scalability in supercomputers. This paper serves as the first step toward building a solid mathematical foundation for this approach. We study the overlapping localized ETD schemes for a model time-dependent diffusion equation discretized in space by the standard central difference. Two methods are proposed and analyzed for solving the fully discrete localized ETD systems: the first one is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second one is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergences of the associated iterative solutions to the corresponding fully discrete localized ETD solution and to the exact semidiscrete solution are rigorously proved. Numerical experiments are also carried out to confirm theoretical results and to compare the performance of the two methods.
exponential time differencing, overlapping domain decomposition, diffusion equation, localization, parallel Schwarz iteration, waveform relaxation
2010 Mathematics Subject Classification
65F60, 65L06, 65M12, 65M55
This work is partially supported by US Department of Energy under grant number DE-SC0016540 and US National Science Foundation under grant number DMS-1521965.
Received 9 December 2017
Accepted 4 April 2018
Published 7 February 2019