论文标题
一种有效的几何方法，用于球体上不可压缩的流体动力学
An efficient geometric method for incompressible hydrodynamics on the sphere
论文作者
论文摘要
我们为球体上的二维理想流体动力学提出了一种有效且高度可扩展的几何方法。起点是Zeitlin的流体动力学有限维模型。效率源于利用离散球形拉普拉斯的三角形分裂，并结合了高度优化的可扩展数值算法。为了进行时间步长时，我们采用了最近开发的等光集成仪，能够保留Euler方程的几何结构，特别是Casimir函数的保护。为了克服以前的计算瓶颈，我们通过一系列三角形特征值问题制定矩阵谎言代数基础，并通过良好成立的线性代数库有效地解决。相同的三角形分裂允许计算流矩阵，涉及逆拉普拉斯式，为此，我们在分布式内存系统上设计了有效的并行实现。所得的总体计算复杂性是$ \ Mathcal {o}（n^3）$ $ n^2 $空间自由度。主导的计算成本是通过并行库Scalapack进行的矩阵矩阵乘法。扩展测试显示，矩阵尺寸$ n = 4096 $的线性比例大约为2500美元$ 2500 $，每个时间阶段的计算时间约为$ 0.55 $秒。这些结果允许长期模拟和收集统计数量，同时保存Casimir功能。我们说明了分辨率$ n = 2048 $的Euler方程的开发算法。
We present an efficient and highly scalable geometric method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a tridiagonal splitting of the discrete spherical Laplacian combined with highly optimized, scalable numerical algorithms. For time-stepping, we adopt a recently developed isospectral integrator able to preserve the geometric structure of Euler's equations, in particular conservation of the Casimir functions. To overcome previous computational bottlenecks, we formulate the matrix Lie algebra basis through a sequence of tridiagonal eigenvalue problems, efficiently solved by well-established linear algebra libraries. The same tridiagonal splitting allows for computation of the stream matrix, involving the inverse Laplacian, for which we design an efficient parallel implementation on distributed memory systems. The resulting overall computational complexity is $\mathcal{O}(N^3)$ per time-step for $N^2$ spatial degrees of freedom. The dominating computational cost is matrix-matrix multiplication, carried out via the parallel library ScaLAPACK. Scaling tests show approximately linear scaling up to around $2500$ cores for the matrix size $N=4096$ with a computational time per time-step of about $0.55$ seconds. These results allow for long-time simulations and the gathering of statistical quantities while simultaneously conserving the Casimir functions. We illustrate the developed algorithm for Euler's equations at the resolution $N=2048$.
