球内Navier-Stokes模拟的最佳复杂度谱方法
An optimal complexity spectral method for Navier--Stokes simulations in the ball
摘要：
本文发展了一种谱方法求解球内不可压广义斯托克斯方程，该方程具有无通量和指定的滑移边界条件。该算法实现了$\mathcal{O}(N\log^2(N))$的每时间步长的最佳复杂度，其中$N$是空间自由度的数量。该方法依赖于螺线管向量场的极线-环面分解、双傅立叶球方法、傅立叶和超球谱方法以及球谐变换来解耦Navier-斯托克斯方程，并达到所需的复杂度和谱精度。
英文摘要：
We develop a spectral method for solving the incompressible generalized Navier--Stokes equations in the ball with no-flux and prescribed slip boundary conditions. The algorithm achieves an optimal complexity per time step of $\mathcal{O}(N\log^2(N))$, where $N$ is the number of spatial degrees of freedom. The method relies on the poloidal-toroidal decomposition of solenoidal vector fields, the double Fourier sphere method, the Fourier and ultraspherical spectral method, and the spherical harmonics transform to decouple the Navier--Stokes equations and achieve the desired complexity and spectral accuracy.
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