混合精度GPU算法：基于Raviart-Thomas有限元的高效湍流模拟
Mixed-precision GPU algorithms for efficient turbulent flow simulations with Raviart-Thomas finite elements
- Ruhr University Bochum（鲁尔大学波鸿）
- Leibniz Supercomputing Centre（莱布尼茨超级计算中心）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出基于Raviart-Thomas有限元的GPU混合精度算法，通过最小二乘投影生成初始猜测，以单精度求解器达到10^{-3}残差容差，在保持湍流统计量的同时实现高达1.7倍加速。
中文摘要 AI 辅助
我们提出了用于不可压缩湍流高保真模拟的GPU算法。空间离散采用H(div)相容的高阶Raviart-Thomas有限元表示速度，压力则采用L^2相容的不连续伽辽金近似。在时间上，使用基于高阶BDF时间步进的一致分裂格式，对流项显式处理。在该格式中，每个时间步需求解一个压力泊松方程和一个关于速度的对称反应扩散型方程。我们开发了一个求解框架，对所有组成部分进行快速的矩阵无关算子求值，并结合用于泊松问题的多重网格求解器，同时提出一个鲁棒的混合精度算法框架。混合精度效率的关键在于使用最小二乘投影为迭代线性求解器生成精确的初始猜测，从而使得相对残差容差可以达到10^{-3}。在此条件下，完全以单精度运行求解器几乎不改变整体迭代次数，并保持关键的湍流统计量，同时相比纯双精度模拟展现出高达1.7倍的加速。
英文摘要
We propose GPU algorithms for high-fidelity simulation of incompressible turbulent flows. Discretization in space is performed with H(div)-conforming high-order Raviart-Thomas finite elements for the velocity and an $L^2$-conforming discontinuous Galerkin approximation for the pressure. In time, a consistent splitting scheme based on higher-order BDF time stepping is used, with convection treated explicitly. In this scheme, a pressure Poisson equation and a symmetric reaction-diffusion-type equation for the velocity need to be solved in each time step. We develop a solution framework with fast matrix-free operator evaluation for all ingredients, combined with multigrid solvers for the Poisson problem, and propose a robust mixed-precision algorithmic framework. A key to mixed-precision efficiency is a least-squares projection to generate accurate initial guesses for the iterative linear solvers, enabling us to work with relative residual tolerances of $10^{-3}$. In this regime, running the solvers entirely in single precision leads to almost no change in overall iteration counts and maintains the crucial turbulence statistics, while showing up to $1.7\times$ speedup over pure double-precision simulations.
