A Hybrid Radial Basis Function - Pseudospectral Method for Thermal Convection in a 3-D Spherical Shell

A novel hybrid spectral method that combines radial basis function (RBF) and Chebyshev pseudospectral (PS) methods in a “2+1” approach is presented for numerically simulating thermal convection in a 3-D spherical shell. This is the first study to apply RBFs to a full 3D physical model in spherical geometry. In addition to being spectrally accurate, RBFs are not defined in terms of any surface based coordinate system such as spherical coordinates. As a result, when used in the lateral directions, as in this study, they completely circumvent the pole issue with the further advantage that nodes can be “scattered” over the surface of a sphere. In the radial direction, Chebyshev polynomials are used, which are also spectrally accurate and provide the necessary clustering near the boundaries to resolve boundary layers. Applications of this new hybrid methodology are given to the problem of convection in the Earth’s mantle,which is modeled by a Boussinesq fluid at infinite Prandtl number. To see whether this numerical technique warrants further investigation, the study limits itself to an isoviscous mantle.Benchmark comparisons are presented with other currently used mantle convection codes for Rayleigh number 7 · 103 and 105. The algorithmic simplicity of the code (mostly due to RBFs)allows it to be written in less than 400 lines of Matlab and run on a single workstation. We find that our method is very competitive with those currently used in the literature

A Fast and Accurate Algorithm for Spherical Harmonic Analysis on HEALPix Grids with Applications to the Cosmic Microwave Background Radiation

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a central component to computing and analyzing the angular power spectrum of the massive CMB data sets. The method uses a novel combination of a non-uniform fast Fourier transform, the double Fourier sphere method, and Slevinsky's fast spherical harmonic transform (Slevinsky, 2019). For a HEALPix grid with $N$ pixels (points), the computational complexity of the method is $\mathcal{O}(N\log^2 N)$, with an initial set-up cost of $\mathcal{O}(N^{3/2}\log N)$. This compares favorably with $\mathcal{O}(N^{3/2})$ runtime complexity of the current methods available in the HEALPix software when multiple maps need to be analyzed at the same time. Using numerical experiments, we demonstrate that the new method also appears to provide better accuracy over the entire angular power spectrum of synthetic data when compared to the current methods, with a convergence rate at least two times higher

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Computing with functions in spherical and polar geometries I. The sphere

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.Comment: 23 page

Stable Computations with Flat Radial Basis Functions Using Vector-Valued Rational Approximations

One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are \u27flat\u27 leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Padé method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson\u27s equation in a 3-D spherical shell