Our understanding of turbulence has heavily relied on high-fidelity Direct Numerical Simulations (DNS) that resolve all dynamically relevant scales. But because of the inherent complexities of turbulent flows, these simulations are computationally very expensive and practically impossible at realistic conditions. Advancements in high performance computing provided much needed boost to the computational resources through increasing levels of parallelism and made DNS realizable, even though only in a limited parameter range. As the number of processing elements (PEs) in parallel machines increases, the penalties incurred in current algorithms due to necessary communications and synchronizations between PEs to update data become significant. These overheads are expected to pose a serious challenge to scalability on the next-generation exascale machines. An effective way to mitigate this bottleneck is through relaxation of strict communication and synchronization constraints and proceed with computations asynchronously i.e. without waiting for updated information from the other PEs. In this work, we investigate the viability of such asynchronous computing using high-order Asynchrony-Tolerant (AT) schemes for accurate and scalable simulations of reacting and non-reacting turbulence at extreme scales. For this, we first assess the important numerical properties of AT schemes, including conservation, stability, and spectral accuracy. Through rigorous mathematical analysis, we expose the breakdown of the standard von Neumann analysis for stability of multi-level schemes, even for widely used synchronous schemes. We overcome these limitations through what we call the generalized von Neumann analysis that is then used to assess stability of the AT schemes. Following which, we propose and implement two computational algorithms to introduce asynchrony in a three-dimensional compressible flow solver. We use these to perform first of a kind asynchronous simulation of compressible turbulence and analyze the effect of asynchrony on important physical characteristics of turbulence. Specifically we show that both large-scale and scale-scale features including highly intermittent instantaneous events, are accurately resolved by these algorithms. We also show excellent strong and weak scaling of asynchronous algorithms up to a processor count of P = 262144 because of significant reduction in communication overheads. As a precursor to the development of asynchronous combustion codes for simulations of more challenging problems with additional physical and numerical complexities, we investigate the effect of asynchrony on several canonical reacting flows. Furthermore, for problems with shocks and discontinuities, such as detonations, we derive and verify AT-WENO (weighted essentially non-oscillatory) schemes. With the ultimate goal to derive new optimal AT schemes we also develop a unified framework for the derivation of finite difference schemes. We show explicit trade-offs between order of accuracy, spectral accuracy and stability under this unifying framework, which can be exploited to devise very accurate numerical schemes for asynchronous computations on extreme scales with minimal overheads
