一种用于大规模并行PDE求解器的异步间断Galerkin方法
An asynchronous discontinuous Galerkin method for massively parallel PDE solvers
AI总结：
本文提出异步DG（ADG）方法，结合异步计算与DG优势，通过放宽通信同步需求提升可扩展性，并引入AT通量恢复精度，数值实验验证其有效性。
摘要：
间断Galerkin（DG）方法因其能够在复杂几何中提供高阶精确解、捕获间断并表现出高算术强度，而被广泛用于求解双曲型偏微分方程（PDE）。然而，基于DG的求解器的可扩展性受到通信瓶颈的阻碍，这些瓶颈源于极端规模下的数据移动和同步需求。为了解决这些挑战，近期研究集中于开发用于PDE求解器的异步计算方法。在此，我们引入了异步DG（ADG）方法，该方法将DG方法的优势与异步计算相结合，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，允许处理单元独立运行，无论通信状态如何，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，并有效解决了由异步性带来的挑战。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格式以及Runge-Kutta（RK）时间积分格式的全离散方程的耗散和色散行为。此外，提出了统计框架内的误差分析，表明使用标准数值通量的ADG方法至多达到一阶精度。为了恢复精度，我们引入了利用多个时间层级数据的异步容忍（AT）通量。进行了广泛的数值实验，以验证ADG-AT格式在线性和非线性问题上的性能。
英文摘要：
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.
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