A unified framework for synchronous and asynchronous optimizedfinite differences for exascale fluid flow simulations.

We present an integrated framework for the development of optimized spatial finite difference schemes that includes formal order of accuracy, spectral resolution and stability. We show how this framework exposes explicitly the tradeoffs between these three elements. This allows us to, for example, construct difference schemes that remain stable for very large time steps, at the expense of spectral accuracy at selected wavenumbers. We extend this framework to derive optimized temporal schemes that exhibit numerical properties similar to schemes with higher order of accuracy. The optimization framework is further extended for the development of asynchronous schemes which mimic the properties of synchronous schemes closely. By relaxing communications, asynchronous schemes provide significant advantages at extreme levels of parallelism and thus provide a suitable path towards effective use of future exascale systems. Asynchronous simulations using asynchrony-tolerant and optimized asynchronous finite difference schemes will be discussed.