Quadrature over the Sphere
نویسندگان
چکیده
Consider integration over the unit sphere in , especially when the integrand has singular behaviour in a polar region. In an earlier paper [4], a numerical integration method was proposed that uses a transformation that leads to an integration problem over the unit sphere with an integrand that is much smoother in the polar regions of the sphere. The transformation uses a grading parameter . The trapezoidal rule is applied to the spherical coordinates representation of the transformed problem. The method is simple to apply, and it was shown in [4] to have convergence or better for integer values of . In this paper, we extend those results to non-integral values of . We also examine superconvergence that was observed when is an odd integer. The overall results agree with those of [11], although the latter is for a different, but related, class of transformations.
منابع مشابه
Thermo-elastic analysis of a functionally graded thick sphere by differential quadrature method
Thermo-elastic analysis of a functionally graded hollow sphere is carried out and numerical solutions of displacement, stress and thermal fields are obtained using the Polynomial differential quadrature (PDQ) method. Material properties are assumed to be graded in the radial direction according to a power law function, ho...
متن کاملOn spherical harmonics based numerical quadrature over the surface of a sphere
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 leastsquares remedy for previously problematic cases. Quadrature accuracies are then compared for different node sets ranging from fully random to those based on Gaussi...
متن کاملبرهمنهی حالتهای همدوس غیرخطی روی سطح کره
In this paper, by using the nonlinear coherent states on a sphere, we introduce superposition of the aforementioned coherent states. Then, we consider quantum optical properties of these new superposed states and compare these properties with the corresponding properties of the nonlinear coherent states on the sphere. Specifically, we investigate their characteristics function, photon-number d...
متن کاملA unified approach to scattered data approximation on S3 and SO(3)
In this paper we use the connection between the rotation group SO(3) and the three-dimensional Euclidean sphere S3 in order to carry over results of the three-sphere directly to the rotation group and vice versa. More precisely, these results connect properties of sampling sets and quadrature formulae on SO(3) and S3, respectively. Furthermore we relate Marcinkiewicz-Zygmund inequalities and co...
متن کاملQuadrature in Besov spaces on the Euclidean sphere
Let q ≥ 1 be an integer, S denote the unit sphere embedded in the Euclidean space Rq+1, and μq be its Lebesgue surface measure. We establish upper and lower bounds for sup f∈B p,ρ ∣∣∣∣ ∫ Sq fdμq − M ∑ k=1 wkf(xk) ∣∣∣∣ , xk ∈ S , wk ∈ R, k = 1, · · · ,M, where B p,ρ is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of xk and wk that admit exact q...
متن کاملNumerical quadrature over the surface of a sphere
Large-scale simulations in spherical geometries require associated quadrature formulas. Classical approaches based on tabulated weights are limited to specific quasi-uniform distributions of relatively low numbers of nodes. By using a radial basis function-generated finite differences (RBF-FD) based approach, the proposed algorithm creates quadrature weights for N arbitrarily scattered nodes in...
متن کامل