A divergence-free spectral expansions method for three-dimensional flows in spherical-gap geometries
Journal of Computational Physics
Calculation of incompressible viscous flows by an unconditionally stable projection FEM
Journal of Computational Physics
Efficient Spectral-Galerkin Methods IV. Spherical Geometries
SIAM Journal on Scientific Computing
Role of the LBB condition in weak spectral projection methods: 405
Journal of Computational Physics
A mixed-basis spectral projection method
Journal of Computational Physics
A Fast and Accurate Numerical Scheme for the Primitive Equations of the Atmosphere
SIAM Journal on Numerical Analysis
Mixed Jacobi-spherical harmonic spectral method for Navier--Stokes equations
Applied Numerical Mathematics
Spectral solvers for spherical elliptic problems
Journal of Computational Physics
Spectral radial basis functions for full sphere computations
Journal of Computational Physics
An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere
Journal of Computational Physics
Hi-index | 31.45 |
The paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere. A variational formulation of the Navier-Stokes equations is adopted using a fractional-step time discretization with the classical second-order backward difference scheme combined with explicit extrapolation of the nonlinear term. The resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors. The spectral matrices for radial operators are characterized by a minimal sparsity - diagonal stiffness and tridiagonal mass matrix. Closed-form expressions of their nonzero elements are provided here for the first time, showing that the condition number of the relevant matrices grows as the second power of the truncation order. A new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the Coriolis force in a rotating frame, but requires a minimal coupling between the modal velocity components in the Fourier space. The numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution, by construction. These results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by Kida and Nakayama, where the velocity field is represented in terms of poloidal and toroidal functions.