High-order asynchrony-tolerant finite difference schemes for partial differential equations

摘要

Synchronizations of processing elements (PEs) in massively parallelsimulations, which arise due to communication or load imbalances between PEs,significantly affect the scalability of scientific applications. We haverecently proposed a method based on finite-difference schemes to solve partialdifferential equations in an asynchronous fashion -- synchronization betweenPEs is relaxed at a mathematical level. While standard schemes can maintaintheir stability in the presence of asynchrony, their accuracy is drasticallyaffected. In this work, we present a general methodology to deriveasynchrony-tolerant (AT) finite difference schemes of arbitrary order ofaccuracy, which can maintain their accuracy when synchronizations are relaxed.We show that there are several choices available in selecting a stencil toderive these schemes and discuss their effect on numerical and computationalperformance. We provide a simple classification of schemes based on the stenciland derive schemes that are representative of different classes. Theirnumerical error is rigorously analyzed within a statistical framework to obtainthe overall accuracy of the solution. Results from numerical experiments areused to validate the performance of the schemes.