Euler-Maclaurin formulas for polynomials on the Cubed Sphere
Abstract
On the sphere, some trapezoidal rule dedicated to the Cubed Sphere grid has been introduced by J.-P.~Croisille about fifteen years ago. As it could be expected, the corresponding quadrature error has an Euler-Maclaurin expansion depending on cross-derivatives at the vertices of a cube and on normal derivatives along the edges. This paper gives a new insight about this expansion. We prove that in fact, for polynomials, the Euler-Maclaurin series under consideration can be expressed in terms of differential operators at the vertices, so that the interior of the edges becomes transparent for the error. Also, an explicit computation of the twelfth-order expansion shows that the order of the suitable differential operators is smaller than the expected ones, which reveals surprising compensations among the error terms. The proposed expansions opens perspectives such as the design of new high-order corrections of the considered trapezoidal rule.
Related
- A unification of Euler-Maclaurin and Euler-Boole summation formulas
- High-order corrections of the trapezoidal rule near the corners of the Cubed Sphere
- A recurrence in logarithmic Möbius sums with coprimality conditions via Euler--Maclaurin
- Block-count constrained harmonic sums: spectral expansion and block-directed Euler-Maclaurin
- Parametric Families of Minimal Positive Cubature Formulas of Degrees 4 and 6 on the 2-Sphere
BibTeX
@misc{bellet2026eulermaclaurin,
title = {{Euler-Maclaurin formulas for polynomials on the Cubed Sphere}},
author = {Jean-Baptiste Bellet},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05739213v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler-Maclaurin formulas for polynomials on the Cubed Sphere
AU - Jean-Baptiste Bellet
PY - 2026
DA - 2026/09/04
PB - HAL
UR - https://hal.science/hal-05739213v1
AB - On the sphere, some trapezoidal rule dedicated to the Cubed Sphere grid has been introduced by J.-P.~Croisille about fifteen years ago. As it could be expected, the corresponding quadrature error has an Euler-Maclaurin expansion depending on cross-derivatives at the vertices of a cube and on normal derivatives along the edges. This paper gives a new insight about this expansion. We prove that in fact, for polynomials, the Euler-Maclaurin series under consideration can be expressed in terms of differential operators at the vertices, so that the interior of the edges becomes transparent for the error. Also, an explicit computation of the twelfth-order expansion shows that the order of the suitable differential operators is smaller than the expected ones, which reveals surprising compensations among the error terms. The proposed expansions opens perspectives such as the design of new high-order corrections of the considered trapezoidal rule.
ER -