A low-storage Runge-Kutta time integration method for scalable asynchrony-tolerant numerical schemes*

Asynchrony-tolerant (AT) finite difference schemes, which relax communication and data synchronization requirements at a mathematical level, are shown to significantly improve the scalability of flow solvers at an extreme scale. These schemes are coupled with suitable high-order time integration methods such as Adams-Bashforth or Runge-Kutta schemes to achieve high-order accurate solutions in time-dependent PDEs. The low-storage RK (LSRK) schemes require low memory compared to the standard schemes and are widely used in several flow solvers. However, these schemes need data communication and synchronization at every time step and at every stage within a time step to achieve high-order accuracy. This work proposes a novel approach to couple AT and LSRK schemes in solving time-dependent PDEs that would significantly reduce communication overheads. The accuracy of this approach is investigated, both theoretically and numerically, using simple 1D model equations. Massively parallel 3D simulations of decaying compressible turbulence are performed to demonstrate the scalability of the proposed approach. At extreme scales, a speed-up of 2.8x was obtained. Overall, the approach shows a promise in addressing the communication bottleneck issue as we move towards exascale.