A spectral method for nonlinear wave equations
Journal of Computational Physics
Fast and accurate spectral treatment of coordinate singularities
Journal of Computational Physics
Spectral collocation methods and polar coordinate singularities
Journal of Computational Physics
Pole condition for singular problems: the pseudospectral approximation
Journal of Computational Physics
Journal of Computational Physics
A pseudospectral approach for polar and spherical geometries
SIAM Journal on Scientific Computing
A spectral method for polar coordinates
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
A spectral method for unbounded domains
Journal of Computational Physics
Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries
SIAM Journal on Scientific Computing
An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries
Journal of Computational Physics
Accurate Navier-Stokes investigation of transitional and turbulent flows in a circular pipe
Journal of Computational Physics
Pseudospectral Solution of the Two-Dimensional Navier--Stokes Equations in a Disk
SIAM Journal on Scientific Computing
Numerical treatment of polar coordinate singularities
Journal of Computational Physics
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A new fast Chebyshev—Fourier algorithm for Poisson-type equations in polar geometries
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
A direct spectral collocation Poisson solver in polar and cylindrical coordinates
Journal of Computational Physics
A MATLAB differentiation matrix suite
ACM Transactions on Mathematical Software (TOMS)
An efficient spectral-projection method for the Navier--Stokes equations in cylindrical geometries
Journal of Computational Physics
Radial Basis Functions
Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Spectral collocation schemes on the unit disc
Journal of Computational Physics
A Fourier-spectral element algorithm for thermal convection in rotating axisymmetric containers
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Pseudo Spectral Methods Applied to Problems in Elasticity
Journal of Scientific Computing
Spectral radial basis functions for full sphere computations
Journal of Computational Physics
Poloidal-toroidal decomposition in a finite cylinder
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Efficient spectral-Galerkin methods for polar and cylindrical geometries
Applied Numerical Mathematics
Efficient implementation of the MFS: The three scenarios
Journal of Computational and Applied Mathematics
Fast radial basis function interpolation with Gaussians by localization and iteration
Journal of Computational Physics
Journal of Computational Physics
A Fourier-Legendre spectral element method in polar coordinates
Journal of Computational Physics
Computers & Mathematics with Applications
Hi-index | 31.46 |
We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier-Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan-Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in @q. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev-Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single ''best'' basis for the disk, but we have laid out the merits and flaws of each spectral option.