Near-algebraic Tchakaloff-like quadrature on spherical triangles
Introduction
In this note we develop an algorithm for the computation of nodes and positive weights of a quadrature formula on spherical triangles, which is nearly exact for algebraic polynomials of a given degree on the 2-sphere, and whose cardinality does not exceed the dimension of the corresponding polynomial space, namely . One of our main goals is also to provide easily usable numerical codes.
Indeed, despite of the relevance of spherical triangles in the field of Geomathematics, the topic of numerical quadrature on spherical triangles, starting from a classical paper by K. Atkinson in the ’80s [1], has received some attention in the literature of the last decades, with however a substantial lack of easily available numerical software (at least to our knowledge); cf. [2], [3], [4], [5] and [6, §7.2] for an overview. Some of the methods have been developed in the framework of numerical PDEs on the sphere, cf. e.g. among others [7], [8]. Notable exceptions are [9], [10], where however the algorithms are not tailored to polynomial spaces, the problem being discussed in the framework of local RBF interpolation on scattered data.
In Section 2 we describe our algorithm (implemented in Matlab) for the construction of the quadrature formula, based on subperiodic trigonometric Gaussian quadrature for planar elliptical sectors [11] and on Caratheodory–Tchakaloff quadrature compression via NNLS [12], and in Section 3 we present some numerical tests.
Section snippets
Quadrature on spherical triangles
We shall concentrate on spherical triangles with centroid at the north pole (with no loss of generality, up to a suitable rotation) that are not “too large” in the sense that they are completely contained in the northern hemisphere, and do not touch the equator. Then we can write the surface integral of a continuous function on such a spherical triangle in Cartesian coordinates (cf. e.g. [13]) where and is
Numerical examples
In this section we present several numerical tests, in order to assess the quality of our compressed near-algebraic quadrature formulas on three spherical triangles with different extension (the key parameter being in (4)). In particular, the triangle of Table 3 is the sphere octant with vertices . In Table 1, Table 2, Table 3 we show the cardinalities of the basic and of the compressed formula, the corresponding compression ratio and the CPU time needed for the
Acknowledgments
Work partially supported by the DOR funds and the biennial project BIRD 192932 of the University of Padova, Italy, and by the INdAM-GNCS . This research has been accomplished within the RITA “Research ITalian network on Approximation” and the UMI Group TAA “Approximation Theory and Applications”.
References (24)
High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere
J. Comput. Phys.
(2006)- et al.
Numerical quadrature over smooth surfaces with boundaries
J. Comput. Phys.
(2018) - et al.
Algebraic cubature by linear blending of elliptical arcs
Appl. Numer. Math.
(2013) - et al.
Polynomial approximation and quadrature on geographic rectangles
Appl. Math. Comput.
(2017) - et al.
Algebraic cubature on polygonal elements with a circular edge
Comput. Math. Appl.
(2020) Numerical integration on the sphere
J. Aust. Math. Soc. Ser. B
(1981)- et al.
Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles
GEM Int. J. Geomath.
(2012) - et al.
Local numerical integration on the sphere
GEM Int. J. Geomath.
(2014) - et al.
Adaptive numerical integration on spherical triangles
Constructing cubature formulae on an arbitrary spherical triangle
J. Fudan Univ. Nat. Sci.
(2006)