E4.11 Localized ETD Based on Domain Decomposition

Abstract

Exponential time differencing (ETD) methods have been proven to be very effective for solving stiff evolution problems in the past decades due to rapid development of computing techniques and capacities. On the other hand, these methods still could become prohibitively expensive for large scale simulations due to the high computational costs for evaluating the products of matrix exponentials and vectors through the Krylov subspace iterative algorithm at each time step. Direct parallelization of the ETD methods is also rarely of good scalability due to the needed global data communication. Therefore, we develop localized ETD methods that use domain decomposition techniques to reduce the size of the problem, in which one instead solves a group of smaller-sized subdomain problems simultaneously with locally computed matrix exponentials. The proposed methods are less expensive and are also more scalable on parallel computers than the classic global ETD methods since only boundary data communication between subdomains is required. We study in algorithm and analysis the localized ETD methods for the time-dependent diffusion problem and the shallow water equations, and some of our progresses and test results will be presented.