'Proxy-equation' paradigm - A novel strategy for massively-parallel asynchronous computations

Massively parallel simulations of transport equation systems call for a paradigm change in algorithm development to achieve efficient scalability. Traditional approaches require time synchronization of processing elements (PEs) which severely restricts scalability. Relaxing synchronization requirement introduces error and slows down convergence. In this paper, we propose and develop a novel `proxy-equation' concept for a general transport equation that (i) tolerates asynchrony with manageable added error, (ii) preserves convergence order and (iii) scales efficiently on massively parallel machines. The central idea is to modify \textit{a priori} the transport equation at the PE boundaries to offset asynchrony errors. Proof-of-concept computations are performed using a one-dimensional advection-diffusion equation. The results demonstrate the promise and advantages of the present strategy.