  Next-generation exascale machines with extreme levels of parallelism will
provide massive computing resources for large scale numerical simulations of
complex physical systems at unprecedented parameter ranges. However, novel
numerical methods, scalable algorithms and re-design of current state-of-the
art numerical solvers are required for scaling to these machines with minimal
overheads. One such approach for partial differential equations based solvers
involves computation of spatial derivatives with possibly delayed or
asynchronous data using high-order asynchrony-tolerant (AT) schemes to
facilitate mitigation of communication and synchronization bottlenecks without
affecting the numerical accuracy. In the present study, an effective
methodology of implementing temporal discretization using a multi-stage
Runge-Kutta method with AT schemes is presented. Together these schemes are
used to perform asynchronous simulations of canonical reacting flow problems,
demonstrated in one-dimension including auto-ignition of a premixture, premixed
flame propagation and non-premixed autoignition. Simulation results show that
the AT schemes incur very small numerical errors in all key quantities of
interest including stiff intermediate species despite delayed data at
processing element (PE) boundaries. For simulations of supersonic flows, the
degraded numerical accuracy of well-known shock-resolving WENO (weighted
essentially non-oscillatory) schemes when used with relaxed synchronization is
also discussed. To overcome this loss of accuracy, high-order AT-WENO schemes
are derived and tested on linear and non-linear equations. Finally the novel
AT-WENO schemes are demonstrated in the propagation of a detonation wave with
delays at PE boundaries.
