An optimal complexity spectral method for Navier–Stokes simulations in the ball
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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 𝒪(Nlog^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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