A scalable asynchronous discontinuous Galerkin method for massively parallel flow simulations

Abstract

Accurate simulations of turbulent flows are crucial forcomprehending complex phenomena in engineered systems and natural processes.These simulations are often computationally expensive and require the use ofsupercomputers, where scalability at extreme scales is significantly affectedby communication overhead. To address this, an asynchronous computing approachfor time-dependent partial differential equations (PDEs) that relaxescommunication/synchronization at a mathematical level has been developed withfinite difference schemes that are ideal for structured meshes. This workproposes an asynchronous discontinuous Galerkin (ADG) method, which has thepotential to provide high-order accurate solutions for various flow problems onboth structured and unstructured meshes, and demonstrates its scalability. Wefirst propose a new method that combines asynchrony-tolerant and low-storageexplicit RK schemes with reduced communication effort. The accuracy of thismethod is assessed both theoretically and numerically, and its scalability isdemonstrated through simulations of the decaying turbulence. Subsequently, weintroduce the asynchronous discontinuous Galerkin method, which combines thebenefits of the DG method with asynchronous computing. The numerical propertiesof the proposed method are investigated, including local conservation,stability, and accuracy, where the method is shown to be, at most, first-orderaccurate. To recover accuracy, we developed new asynchrony-tolerant (AT) fluxesthat utilize data from multiple time levels. To validate these theoreticalfindings, several numerical experiments are conducted based on both linear andnonlinear problems. Finally, we develop a parallel PDE solver based on the ADGmethod within an open-source finite element library deal.II using acommunication-avoiding algorithm. Accuracy validation and scalabilitybenchmarks of the solver are performed, demonstrating a speedup of up to 80%with the ADG method at an extreme scale with 9216 cores. The overall workhighlights the potential benefits of the asynchronous approach for the developmentof accurate and scalable PDE solvers, paving the way for simulations of complexphysical systems on massively parallel supercomputers.

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