The discontinuous Galerkin (DG) method is widely being used to solve hyperbolic partial differential equations (PDEs) due to its ability to provide high-order accurate solutions in complex geometries, capture discontinuities, and exhibit high arithmetic intensity. However, the scalability of DG-based solvers is impeded by communication bottlenecks arising from the data movement and synchronization requirements at extreme scales. To address these challenges, recent studies have focused on the development of asynchronous computing approaches for PDE solvers. Herein, we introduce the asynchronous DG (ADG) method, which combines the benefits of the DG method with asynchronous computing to overcome communication bottlenecks. The ADG method relaxes the need for data communication and synchronization at a mathematical level, allowing processing elements to operate independently regardless of the communication status, thus potentially improving the scalability of solvers. The proposed ADG method ensures flux conservation and effectively addresses challenges arising from asynchrony. To assess its stability, Fourier-mode analysis is employed to examine the dissipation and dispersion behavior of fully-discrete equations that use the DG and ADG schemes along with the Runge–Kutta (RK) time integration scheme. Furthermore, an error analysis within a statistical framework is presented, which demonstrates that the ADG method with standard numerical fluxes achieves at most first-order accuracy. To recover accuracy, we introduce asynchrony-tolerant (AT) fluxes that utilize data from multiple time levels. Extensive numerical experiments were conducted to validate the performance of the ADG-AT scheme for both linear and nonlinear problems. Overall, the proposed ADG-AT method demonstrates the potential to achieve accurate and scalable DG-based PDE solvers, paving the way for simulations of complex physical systems on massively parallel supercomputers.

中文翻译：

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大规模并行 PDE 求解器的异步间断 Galerkin 方法


间断伽辽金（DG）方法因其能够在复杂几何形状中提供高阶精确解、捕获不连续性并表现出高算术强度而被广泛用于求解双曲偏微分方程（PDE）。然而，基于 DG 的求解器的可扩展性受到极端规模的数据移动和同步要求所产生的通信瓶颈的阻碍。为了应对这些挑战，最近的研究集中在偏微分方程求解器的异步计算方法的开发上。在这里，我们介绍异步DG（ADG）方法，它将DG方法的优点与异步计算相结合，以克服通信瓶颈。 ADG方法在数学层面上放宽了对数据通信和同步的需求，允许处理元件独立操作而不管通信状态如何，从而潜在地提高求解器的可扩展性。所提出的 ADG 方法确保了通量守恒，并有效解决了异步带来的挑战。为了评估其稳定性，采用傅里叶模式分析来检查使用 DG 和 ADG 方案以及龙格-库塔 (RK) 时间积分方案的全离散方程的耗散和色散行为。此外，还提出了统计框架内的误差分析，表明具有标准数值通量的 ADG 方法最多可实现一阶精度。为了恢复准确性，我们引入了利用多个时间级别的数据的异步容忍（AT）通量。进行了大量的数值实验来验证 ADG-AT 方案对于线性和非线性问题的性能。 总体而言，所提出的 ADG-AT 方法展示了实现精确且可扩展的基于 DG 的 PDE 求解器的潜力，为大规模并行超级计算机上复杂物理系统的模拟铺平了道路。