  Direct numerical simulations (DNS) are an indispensable tool for
understanding the fundamental physics of turbulent flows. Because of their
steep increase in computational cost with Reynolds number ($R_{\lambda}$),
well-resolved DNS are realizable only on massively parallel supercomputers,
even at moderate $R_{\lambda}$. However, at extreme scales, the communications
and synchronizations between processing elements (PEs) involved in current
approaches become exceedingly expensive and are expected to be a major
bottleneck to scalability. In order to overcome this challenge, we developed
algorithms using the so-called Asynchrony-Tolerant (AT) schemes that relax
communication and synchronization constraints at a mathematical level, to
perform DNS of decaying and solenoidally forced compressible turbulence.
Asynchrony is introduced using two approaches, one that avoids synchronizations
and the other that avoids communications. These result in periodic and random
delays, respectively, at PE boundaries. We show that both asynchronous
algorithms accurately resolve the large-scale and small-scale motions of
turbulence, including instantaneous and intermittent fields. We also show that
in asynchronous simulations the communication time is a relatively smaller
fraction of the total computation time, especially at large processor count,
compared to standard synchronous simulations. As a consequence, we observe
improved parallel scalability up to $262144$ processors for both asynchronous
algorithms.
