Vector spherical harmonics on @math have wide applications in geophysics, quantum mechanics and astrophysics. In the representation of a tangent field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this paper, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms for these evaluations. The forward FaVeST which evaluates the Fourier coefficients has computational steps proportional to @math for @math number of evaluation points. The adjoint FaVeST which evaluates a linear combination of vector spherical harmonics with degree up to @math for @math evaluation points is proportional to @math . Numerical examples illustrate the accuracy and efficiency of FaVeST.
In contrast, fast transforms for vector spherical harmonics receive less attention. To the best of our knowledge, there are no existing fast algorithms for the forward and adjoint vector spherical harmonic transforms. @cite_58 simply use FFTs to speed up their algorithms for solving Navier-Stokes PDEs on the unit sphere. However, their method is based on the idea that applies conventional FFTs to evaluate complex azimuthal exponential terms involved in the formulation of vector spherical harmonics. As fast Legendre transforms are not implemented, their method is not a fast transform. @cite_53 evaluate vector spherical harmonic expansions via spectral element grids, which is neither fast computation.
@cite_53 @cite_58
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We present a Fourier continuation (FC) algorithm for the solution of the fully nonlinear compressible Navier-Stokes equations in general spatial domains. The new scheme is based on the recently introduced accelerated FC method, which enables use of highly accurate Fourier expansions as the main building block of general-domain PDE solvers. Previous FC-based PDE solvers are restricted to linear scalar equations with constant coefficients. The FC methodology presented in this text thus constitutes a significant generalization of the previous FC schemes, as it yields general-domain FC solvers for nonlinear systems of PDEs. While not restricted to periodic boundary conditions and therefore applicable to general boundary value problems on arbitrary domains, the proposed algorithm inherits many of the highly desirable properties arising from rapidly convergent Fourier expansions, including high-order convergence, essentially spectrally accurate dispersion relations, and much milder CFL constraints than those imposed by polynomial-based spectral methods-since, for example, the spectral radius of the FC first derivative grows linearly with the number of spatial discretization points. We demonstrate the accuracy and optimal parallel efficiency of the algorithm in a variety of scientific and engineering contexts relevant to fluid-dynamics and nonlinear acoustics. The spectral transform method is a standard numerical technique for solving partial differential equations on a sphere and is widely used in atmospheric circulation models. Recent research has identified several promising algorithms for implementing this method on massively parallel computers; however, no detailed comparison of the different algorithms has previously been attempted. In this paper, we describe these different parallel algorithms and report on computational experiments that we have conducted to evaluate their efficiency on parallel computers. The experiments used a testbed code that solves the nonlinear shallow water equations on a sphere; considerable care was taken to ensure that the experiments provide a fair comparison of the different algorithms and that the results are relevant to global models. We focus on hypercube- and mesh-connected multicomputers with cut-through routing, such as the Intel iPSC 860, DELTA, and Paragon, and the nCUBE 2, but we also indicate how the results extend to other parallel computer architectures. The results of this study are relevant not only to the spectral transform method but also to multidimensional fast Fourier transforms (FFTs) and other parallel transforms. A group of algorithms is presented generalizing the fast Fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval @math . The schemes of this paper are based on a combination of certain analytical considerations with the classical fast Fourier transform and generalize both the forward and backward FFTs. Each of the algorithms requires @math arithmetic operations, where @math is the precision of computations and N is the number of nodes. The efficiency of the approach is illustrated by several numerical examples. Abstract In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N , the proposed algorithm's complexity is O ( N 2 log N ) , just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT's and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT's, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT's and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.