Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

Overlapping localized exponential time differencing methods for diffusion problems

Pages: 1531 – 1555

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n6.a3

Authors

Thi-Thao-Phuong Hoang (Department of Mathematics and Interdisciplinary Mathematics Institute, University of South Carolina, Columbia, S.C., U.S.A.)

Lili Ju (Department of Mathematics and Interdisciplinary Mathematics Institute, University of South Carolina, Columbia, S.C., U.S.A.)

Zhu Wang (Department of Mathematics and Interdisciplinary Mathematics Institute, University of South Carolina, Columbia, S.C., U.S.A.)

Abstract

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.

Keywords

exponential time differencing, overlapping domain decomposition, diffusion equation, localization, parallel Schwarz iteration, waveform relaxation

2010 Mathematics Subject Classification

65F60, 65L06, 65M12, 65M55

Full Text (PDF format)

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