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Barycentric interpolation formulas for the sphere and the disk

Michael Chiwere, Grady B. Wright

Abstract

Spherical and polar geometries arise in many important areas of computational science, including weather and climate forecasting, optics, and astrophysics. In these applications, tensor-product grids are often used to represent unknowns. However, interpolation schemes that exploit the tensor-product structure can introduce artificial boundaries at the poles in spherical coordinates and at the origin in polar coordinates, leading to numerical challenges, especially for high-order methods. In this paper, we present new bivariate trigonometric barycentric interpolation formulas for spheres and bivariate trigonometric/polynomial barycentric formulas for disks, designed to overcome these issues. These formulas are also efficient, as they only rely on a set of (precomputed) weights that depend on the grid structure and not the data itself. The formulas are based on the Double Fourier Sphere (DFS) method, which transforms the sphere into a doubly periodic domain and the disk into a domain without an artificial boundary at the origin. For standard tensor-product grids, the proposed formulas exhibit exponential convergence when approximating smooth functions. We provide numerical results to demonstrate these convergence rates and showcase an application of the spherical barycentric formulas in a semi-Lagrangian advection scheme for solving the tracer transport equation on the sphere.

Barycentric interpolation formulas for the sphere and the disk

Abstract

Spherical and polar geometries arise in many important areas of computational science, including weather and climate forecasting, optics, and astrophysics. In these applications, tensor-product grids are often used to represent unknowns. However, interpolation schemes that exploit the tensor-product structure can introduce artificial boundaries at the poles in spherical coordinates and at the origin in polar coordinates, leading to numerical challenges, especially for high-order methods. In this paper, we present new bivariate trigonometric barycentric interpolation formulas for spheres and bivariate trigonometric/polynomial barycentric formulas for disks, designed to overcome these issues. These formulas are also efficient, as they only rely on a set of (precomputed) weights that depend on the grid structure and not the data itself. The formulas are based on the Double Fourier Sphere (DFS) method, which transforms the sphere into a doubly periodic domain and the disk into a domain without an artificial boundary at the origin. For standard tensor-product grids, the proposed formulas exhibit exponential convergence when approximating smooth functions. We provide numerical results to demonstrate these convergence rates and showcase an application of the spherical barycentric formulas in a semi-Lagrangian advection scheme for solving the tracer transport equation on the sphere.
Paper Structure (17 sections, 4 theorems, 38 equations, 7 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 4 theorems, 38 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Double Fourier Sphere (DFS) method
  3. BMC symmetry and even-periodic/odd-antiperiodic decompositions
  4. Barycentric interpolation formulas for the sphere
  5. Common tensor product grids
  6. Barycentric formulas for an even or odd periodic function
  7. Barycentric formulas for a $\pi$-periodic or $\pi$-antiperiodic function
  8. Barycentric formulas for the sphere
  9. Numerical example
  10. Barycentric interpolation formulas for the disk
  11. Common tensor product grids
  12. Barycentric formulas for an even or odd function
  13. Barycentric formulas for the disk
  14. Numerical example
  15. Semi-Lagrangian advection (SLA) on the sphere
  16. ...and 2 more sections

Key Result

Lemma 1

Let $f:[0,2\pi]\times[0,\pi]\rightarrow \mathbb{C}$ have the additive decomposition where $f^-(\phi , \theta) = \frac{1}{2}\left(g(\phi,\theta)-h(\phi , \theta)\right)$ and $f^+(\phi , \theta) = \frac{1}{2}\left(g(\phi, \theta) + h(\phi, \theta)\right)$, with $g$ and $h$ taking the same definitions as in eq:ftilde_sph. If $f^-$ is odd, $2\pi$-periodic in $\theta$, and $\pi-$antiper

Figures (7)

  • Figure 1: The DFS method applied to a football (or soccer ball) pattern on the sphere. (a) Football pattern---note the rotation to avoid symmetries with respect to the standard coordinate axes. (b) The projection of the football pattern using spherical coordinates. (c) Football pattern after applying the DFS method resulting in a doubly periodic pattern.
  • Figure 2: Illustrations of common spherical grids for which the barycentric formulas are applicable. For the EQ grid $m=n-1=9$, while for the SEQ and GL grids $m=n=9$. Grid points are marked by blue dots and the solid lines correspond to an EQ grid for reference.
  • Figure 3: (a) Test function \ref{['eq:funs']} on the sphere, where dark blue corresponds to -1 and bright yellow to 1. (b) Relative max-norm error in the barycentric interpolants \ref{['eq:bary_sphere']} of the function in (a) for the different grids using $n = m$.
  • Figure 4: Comparison of Fouier-Chebyshev grids, where the points in the radial direction are (a) the Chebyshev points (of the second kind) defined over $[0,1]$ and (b) the Chebyshev points (of the second kind) defined over $[-1,1]$ and restricted to $[0,1]$ as in the CH2 definition. Both grids have the same number of points in the polar and radial directions.
  • Figure 5: (a) Test function \ref{['eq:fund']} on the disk, where dark blue is corresponds to -1 and bright yellow to 1. (b) Relative max-norm error in the barycentric interpolants \ref{['eq:bary_disk']} of the function in (a) for the CH1, CH2, and GL grids using $n = m$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2