A spectral method for nonlocal diffusion operators on the sphere

Slevinsky, Richard M.; Montanelli, Hadrien; Du, Qiang

doi:10.1016/j.jcp.2018.06.024

Cited by 15 publications

(23 citation statements)

References 44 publications

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“…At the moment, the theory on AC schemes does not offer any estimate of the order of convergence with respect to different couplings of h and δ. Preliminary numerical experiments in offer some insight about the balance of modelling and discretization errors, but additional theoretical analyses need to be carried out, except for the case of Fourier spectral approximations of nonlocal models with periodic boundary conditions, for which precise error estimates can be found in Du andYang (2016, 2017) and Slevinsky, Montanelli and Du (2018).…”

Section: Asymptotically Compatible Conforming Finite Element Methods mentioning

confidence: 99%

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Numerical methods for nonlocal and fractional models

et al. 2020

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Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

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Section: Asymptotically Compatible Conforming Finite Element Methods mentioning

confidence: 99%

“…As a result, K(nδ) can be accurately evaluated for large nδ by using, for example, a high-order Runge-Kutta method. Slevinsky et al (2018) proposed a Fourier spectral method to solve nonlocal diffusion models on the unit sphere S 2 ⊂ R 3 , where the nonlocal Laplace-Beltrami operator is defined by…”

Section: Spectral-galerkin Methods For Nonlocal Diffusionmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…The unit ball has three separate coordinate singularities: disk-like singularities at the north and south pole, and a third at the centre of the domain. Spherical geometry is important for a large number of two-and three-dimensional applications; see e.g., [1,46,50,43,42,48,29,17,22]. Astrophysical and planetary applications provide many obvious examples with stars and planets.…”

Section: Introductionmentioning

confidence: 99%

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

Vasil

Lecoanet

Burns

et al. 2019

Journal of Computational Physics: X

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This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spinweighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as grad and div. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

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“…where B δ (x x x) = {y y y ∈ R d : |x x x − y y y| δ } is the closed ball of radius 0 < δ < ∞ centred at x x x and the kernel ρ δ is a nonnegative spherically symmetric function of the euclidean distance compactly supported on [0, δ ]. Nonlocal diffusion has also been analyzed in different geometries, including the real line by Zheng et al (2017) and the unit 2-sphere by Slevinsky et al (2018). In 1, 2, and 3 spatial dimensions, Du & Yang (2017) have developed algorithms for the numerical evaluation of nonlocal diffusion of Fourier series.…”

Section: Introductionmentioning

confidence: 99%

Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices

Slevinsky

2019

Journal of Computational Physics

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Received on ; revised on ]We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source and target tends to 0. Rapid algorithms are derived for their Fourier spectra with the computation of each eigenvalue independent of all others. The algorithms are trivially parallelizable, capable of leveraging more powerful compute environments, and the accuracy of the eigenvalues is individually controllable. The algorithms include a Maclaurin series and a full divergent asymptotic series valid for any d spatial dimensions. Using Drummond's sequence transformation, we prove linear complexity recurrence relations for degree-graded sequences of numerators and denominators in the rational approximations to the divergent asymptotic series. These relations are important to ensure that the algorithms are efficient, and also increase the numerical stability compared with the conventional algorithm with quadratic complexity.

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A spectral method for nonlocal diffusion operators on the sphere

Cited by 15 publications

References 44 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations

Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices

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