Advances in Computational Mathematics

, Volume 40, Issue 5–6, pp 1169–1184 | Cite as

On spherical harmonics based numerical quadrature over the surface of a sphere

  • Bengt FornbergEmail author
  • Jordan M. Martel


It has been suggested in the literature that different quasi-uniform node sets on a sphere lead to quadrature formulas of highly variable quality. We analyze here the nature of these variations, and describe an easy-to-implement least-squares remedy for previously problematic cases. Quadrature accuracies are then compared for different node sets ranging from fully random to those based on Gaussian quadrature concepts.


Quadrature Sphere Spherical harmonics ME nodes MD nodes Gaussian quadrature 

Mathematics Subject Classifications (2010)

65D30 65D32 41A55 


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  1. 1.
    Ahrens, C., Beylkin, G.: Rotationally invariant quadratures for the sphere. Proc. Roy. Soc. A 465, 3103–3125 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2000)Google Scholar
  3. 3.
    Curtis, P.C. Jr.: n-parameter families and best approximation. Pac. J. Math. 93, 1013–1027 (1959)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Flyer, N., Fornberg, B.: Radial basis functions: Developments and applications to planetary scale flows. Comput. Fluids 46, 23–32 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Flyer, N., Lehto, E., Blaise, S., Wright, G.B., St-Cyr, A.: A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. J. Comput. Phys. 231, 4078–4095 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Flyer, N., Wright, G.B., Fornberg, B.: Radial basis function-generated finite differences: A mesh-free method for computational geosciences. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics. Springer, Berlin (2014)Google Scholar
  7. 7.
    Fornberg, B., Lehto, E.: Stabilization of RBF-generated finite difference methods for convective PDEsJ. Comput. Phys. 230, 2270–2285 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fornberg, B., Piret, C.: On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere. J. Comput. Phys. 227, 2758–2780 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2(1), 84–90 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hesse, K., Sloan, I.H., Womersley, R.S.: Numerical integration on the sphere. In: Freeden,W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 11871219. Springer, Berlin (2010)Google Scholar
  11. 11.
    Mairhuber, J.C.: On Haar’s theorem concerning Chebyshev approximation problems having unique solutions. Proc. Amer. Math. Soc. 7(4), 609–615 (1956)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Manuel, G., Kunis, S., Potts, D.: On the computation of nonnegative quadrature weights on the sphere. Appl. Comp. Harm. Anal. 27(1), 124–132 (2009)CrossRefzbMATHGoogle Scholar
  13. 13.
    McLaren, A.D.: Optimal numerical integration on a sphere. Math. Comput. 17(84), 361–383 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Mhaskar, H., Narcowich, F., Ward, J.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comput. 70(235), 1113–1130 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Renka, R.J.: Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Softw. 14, 139–148 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Sommariva, A., Womersley, R.S.: Integration by RBF over the Sphere. Mathematics Report AMR 05/17. University of New South Wales (2005)Google Scholar
  17. 17.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50(1), 67–87 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Womersley, R.S., Sloan, I.H.: How good can polynomial interpolation on the sphere be? Adv. Comput. Math. 23, 195–226 (2001)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Womersley, R. S., Sloan, I.H.: Interpolation and Cubature on the Sphere. (2003)

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Colorado at BoulderBoulderUSA
  2. 2.Leeds School of BusinessUniversity of Colorado at BoulderBoulderUSA

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