An optimal complexity spectral method for Navier--Stokes simulations in the ball

Abstract

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 ( log 2 ( )), where is the number of spatial degrees of freedom. The method relies on the poloidaltoroidal 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.

Approximation properties of the double Fourier sphere method

“…Proof. We note that φ l is the product of sine or cosine for any l ∈ [3]. Therefore, all partial derivatives are also the product of sine, cosine or zero, which implies (15).…”

“…Sampling methods on the sphere based on spherical harmonics and the DFS method were compared in [29]. Recently, the DFS method was utilized in the numerical solution of partial differential equations on the sphere [25] and the ball [4,3] as well as in computational spherical harmonics analysis [9]. To the best of our knowledge, no results on the convergence of the approximation with the DFS method were published so far.…”

A double Fourier sphere method for $d$-dimensional manifolds

“…e.g., [11,19,27,45]. The problem of representing and numerically manipulating such functions arises in various areas as application including weather prediction [8,9,29], protein docking [37], active fluids in biology [2], geosciences [20,23], and astrophysics [21,41].…”

Approximation Properties of the Double Fourier Sphere Method

“…Recently, the DFS method was utilized in the numerical solution of partial differential equations on the sphere [28] and in computational spherical harmonics analysis [11]. Moreover, it can be generalized to other geometries such as the unit disk [42], the ball [4,5], and the cylinder [15]. To the best of our knowledge, no results on the convergence of the approximation with the DFS method have been published so far.…”

“…Fig 4. Logarithmic plot of the maximal error f − R h f C(S 2 ) of the DFS Fourier sum (blue) and the maximal error of the truncated spherical harmonics series (red, dashed), depending on the degree h. While the error behavior is similar, the DFS Fourier expansion can be computed much faster…”