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

## TL;DR

This paper develops high-order finite difference schemes that remain accurate despite asynchronous processing in parallel PDE simulations, improving scalability without sacrificing precision.

## Contribution

It introduces a general methodology for deriving high-order asynchrony-tolerant finite difference schemes that preserve accuracy under relaxed synchronization.

## Key findings

- Schemes maintain stability and accuracy with asynchrony
- Numerical error analyzed within a statistical framework
- Validated performance through numerical experiments

## Abstract

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.