用于大气流动模拟的GPU加速嵌套舒尔补求解器——采用水平显式、垂直隐式格式与间断伽辽金方法
GPU-Accelerated Nested Schur-Complement Solvers for Atmospheric Flow Simulations Using Horizontally-Explicit, Vertically-Implicit Schemes and Discontinuous Galerkin Methods
AI总结：
本研究针对带重力的可压缩欧拉方程，提出GPU加速嵌套舒尔补求解器，结合水平显式垂直隐式格式与间断伽辽金方法，实现高效大气流动模拟，在NVIDIA GH200 GPU上获高模拟效率与鲁棒性。
摘要：
水平显式、垂直隐式（HEVI）时间积分可消除非静力大气模型中细垂直网格带来的声波时间步长限制，但需要快速求解所得的垂直线性系统，尤其是在图形处理器（GPU）上。针对应用于带重力的可压缩欧拉方程的勒让德-高斯-洛巴托（LGL）节点上的分裂形式间断伽辽金谱元法，我们提出了用于这些系统的GPU加速嵌套舒尔补求解器。时间上，我们采用罗森布罗克-威（Rosenbrock-W）方法，这是一种线性隐式龙格-库塔格式，使用近似雅可比矩阵时仍能保持其阶数和稳定性。这允许生成稀疏垂直雅可比矩阵，该矩阵可捕捉带地形的曲线网格上的垂直声波和重力，并解耦为独立的柱系统。求解器是嵌套的：全局舒尔补（静态凝聚）利用LGL稀疏性将每个柱凝聚到几个单元边未知量上，而局部舒尔补则消除每个单元内除垂直动量外的所有变量，利用压力仅依赖位温密度的特性。我们提出并分析了两种凝聚策略，它们分别产生带状和块三对角约化系统。所得HEVI格式可将总质量和位温密度守恒至舍入误差水平。求解器在this http URL（this http URL的大气扩展版本）中针对中央处理器（CPU）和GPU实现。我们通过二维通道中的线性惯性重力波验证了高阶收敛性，并在球体上针对斜压不稳定、地形诱发的斜压不稳定以及DCMIP-2025间隙流与涡旋脱落案例展示了其鲁棒性。对于无地形的斜压不稳定，该格式在单块NVIDIA GH200 GPU上以双精度运行时，在100公里和50公里分辨率下分别达到每天18.8和2.9个模拟年。
英文摘要：
Horizontally explicit, vertically implicit (HEVI) time integration removes the acoustic time-step restriction imposed by fine vertical grids in nonhydrostatic atmospheric models, but requires fast solvers for the resulting vertical linear systems, especially on GPUs. We present GPU-accelerated nested Schur-complement solvers for these systems for a split-form discontinuous Galerkin spectral element method on Legendre-Gauss-Lobatto (LGL) nodes applied to the compressible Euler equations with gravity. In time, we use a Rosenbrock-W method, a linearly implicit Runge-Kutta scheme that retains its order and stability with approximate Jacobians. This allows a sparse vertical Jacobian that captures vertical acoustic waves and gravity on curvilinear meshes with orography and decouples into independent column systems. The solvers are nested: a global Schur complement (static condensation) exploits the LGL sparsity to condense each column onto a few element-edge unknowns, and a local Schur complement eliminates all variables but the vertical momentum within each element, exploiting that the pressure depends only on the potential-temperature density. We propose and analyze two condensation strategies, which lead to banded and block-tridiagonal reduced systems. The resulting HEVI schemes conserve total mass and potential-temperature density to round-off. The solvers are implemented for CPUs and GPUs in TrixiAtmo.jl, the atmospheric extension of Trixi.jl. We verify high-order convergence with linear inertia-gravity waves in a 2D channel and demonstrate robustness on the sphere with the baroclinic instability, a mountain-induced baroclinic instability, and the DCMIP-2025 gap-flow and vortex-shedding cases. For the baroclinic instability without orography, the scheme achieves 18.8 and 2.9 simulated years per day at 100 and 50 km resolution on one NVIDIA GH200 GPU in double precision.
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