  Synchronization overheads pose a major challenge as applications advance
towards extreme scales. In current large-scale algorithms, synchronization as
well as data communication delay the parallel computations at each time step in
a time-dependent partial differential equation (PDE) solver. This creates a new
scaling wall when moving towards exascale. We present a weakly-synchronous
algorithm based on novel asynchrony-tolerant (AT) finite-difference schemes
that relax synchronization at a mathematical level. We utilize remote memory
access programming schemes that have been shown to provide significant speedup
on modern supercomputers, to efficiently implement communications suitable for
AT schemes, and compare to two-sided communications that are state-of-practice.
We present results from simulations of Burgers' equation as a model of
multi-scale strongly non-linear dynamical systems. Our algorithm demonstrate
excellent scalability of the new AT schemes for large-scale computing, with a
speedup of up to $3.3$x in communication time and $2.19$x in total runtime. We
expect that such schemes can form the basis for exascale PDE algorithms.
