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

Nicolas Boullé, Jonasz Słomka, Alex Townsend

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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