RBF based quadrature on the sphere

• Autorzy: Kindelan Manuel , González-Rodríguez Pedro , Álvarez Diego

Identyfikatory

DOI

10.34768/amcs-2022-0034

• Języki publikacji: EN

Abstrakty

EN

The paper describes a new RBF-FD based technique to compute quadrature weights on the sphere. In the proposed method, the sphere is divided into rectangles in the latitude-azimuth coordinate system, and the function is integrated over each rectangle using RBF interpolation. The method is easy to implement and its accuracy is comparable to that based on SPH expansions. One advantage of the proposed method is its ability to handle non-uniform node distributions. On this respect, we propose a new algorithm to cluster nodes in regions of steep changes in the function. It is a repulsion-based algorithm with a non-uniform distribution of electrical charges. We show that, using node clustering, the accuracy of the method can be significantly improved.

Słowa kluczowe

EN

RBF-FD based technique; quadrature; sphere

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2022
• Tom: Vol. 32, no. 3
• Strony: 467--479
• Opis fizyczny: Bibliogr. 24 poz., rys., tab., wykr.

Twórcy

autor

Kindelan Manuel

• Department of Mathematics, Carlos III University of Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

autor

González-Rodríguez Pedro

• Department of Mathematics, Carlos III University of Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

autor

Álvarez Diego

• Department of Mathematics, Carlos III University of Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain

Bibliografia

• [1] Ahrens, C. and Beylkin, G. (2009). Rotationally invariant quadratures for the sphere, Proceedings of the Royal Society A 465(2110): 3103–3125.
• [2] Atkinson, K. (1982). Numerical integration on the sphere, Journal of the Australian Mathematical Society B 23(3): 332–347.
• [3] Bazant, Z. and Oh, B. (1986). Efficient numerical integration on the surface of a sphere, Journal of Applied Mathematics and Mechanics 66(11): 37–49.
• [4] Beentjes, C. (2015). Quadrature on a spherical surface, https://cbeentjes.github.io/files/Ramblings/QuadratureSphere.pdf.
• [5] Bruno, O.P. and Kunyansky, L.A. (2001). A fast, high order algorithm for the solution of surface scattering problems: Basic implementation, tests and applications, Journal of Computational Physics 169(1): 80–110.
• [6] Flyer, N. and Fornberg, B. (2011). Radial basis functions: Developments and applications to planetary scale flows, Computers and Fluids 46(1): 23–32.
• [7] Flyer, N., Lehto, E., Blaise, S.,Wright, G. and St-Cyr, A. (2012). A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere, Journal of Computational Physics 231(11): 4078–4095.
• [8] Flyer, N., Wright, G. and Fornberg, B. (2014). Radial basis function generated finite differences: A mesh-free method for computational geosciences, in M. Freeden et al. (Eds), Handbook of Geomathematics, Springer-Verlag, Berlin, pp. 2535–2669.
• [9] Fornberg, B. and Flyer, N. (2015). A Primer on Radial Basis Functions with Applications to the Geosciences, Society for Industrial and Applied Mathematics, Philadelphia.
• [10] Fornberg, B. and Martel, J. (2014). On spherical harmonics based numerical quadrature over the surface of a sphere, Advances in Computational Mathematics 40(5–6): 1169–1184.
• [11] Fornberg, B. and Piret, C. (2007). A stable algorithm for flat radial basis function on a sphere, SIAM Journal on Scientific Computing 30(1): 60–80.
• [12] Fornberg, B. and Piret, C. (2008). On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere, Journal of Computational Physics 227(5): 2758–2780.
• [13] Fuselier, E., Hangelbroek, T., Narcowich, F., Ward, J. and Wright, B. (2014). Kernel based quadrature on spheres and other homogeneous spaces, Numerische Mathematik 127(1): 57–92.
• [14] Halton, J. (1960). On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numerische Mathematik 1(1): 84–90.
• [15] Hesse, K., Sloan, I. and Womersley, R. (2010). Numerical integration on the sphere, in M. Freeden et al. (Eds), Handbook of Geomathematics, Springer-Verlag, Berlin, pp. 1187–1219.
• [16] Klöckner, A., Barnett, A., Greengard, L. and O’Neil, M. (2013). Quadrature by expansion: A new method for the evaluation of layer potentials, Journal of Computational Physics 252: 332–349.
• [17] Klinteberg, L. and Tornberg, A.K. (2016). A fast integral equation method for solid particles in viscous flow using quadrature by expansion, Journal of Computational Physics 326: 420–445.
• [18] Reeger, J. (2015). Spherical Quadrature RBF (Quadrature Nodes), https://es.mathworks.com/matlabcentral/fileexchange/51214.
• [19] Reeger, J. and Fornberg, B. (2016). Numerical quadrature over the surface of a sphere, Studies in Applied Mathematics 137(1): 174–188.
• [20] Reeger, J., Fornberg, B. and Watts, L. (2016). Numerical quadrature over smooth, closed surfaces, Proceedings of the Royal Society A 472(2194): 20160401.
• [21] Saff, E. and Kuijlaar, A. (1997). Distributing many points on a sphere, Proceedings of the Royal Society A 19(2): 5–11.
• [22] Sommariva, A. and Womersley, R. (2005). Integration by RBF over the sphere, Applied Mathematics Report amr05/17, University of New South Wales, Sydney.
• [23] Stroud, A. (1971). Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs.
• [24] Womersley, R. and Sloan, H. (2003). Interpolation and cubature on the sphere, https://web.maths.unsw.edu.au/˜rsw/Sphere/.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)