High-order asynchrony-tolerant finite difference schemes for partial differential equations
  Synchronizations of processing elements (PEs) in massively parallel
simulations, which arise due to communication or load imbalances between PEs,
significantly affect the scalability of scientific applications. We have
recently proposed a method based on finite-difference schemes to solve partial
differential equations in an asynchronous fashion -- synchronization between
PEs is relaxed at a mathematical level. While standard schemes can maintain
their stability in the presence of asynchrony, their accuracy is drastically
affected. In this work, we present a general methodology to derive
asynchrony-tolerant (AT) finite difference schemes of arbitrary order of
accuracy, which can maintain their accuracy when synchronizations are relaxed.
We show that there are several choices available in selecting a stencil to
derive these schemes and discuss their effect on numerical and computational
performance. We provide a simple classification of schemes based on the stencil
and derive schemes that are representative of different classes. Their
numerical error is rigorously analyzed within a statistical framework to obtain
the overall accuracy of the solution. Results from numerical experiments are
used to validate the performance of the schemes.
