High-order corrections of the trapezoidal rule near the corners of the Cubed Sphere
Abstract
The trapezoidal rule allows to define a simple spherical quadrature rule on the Cubed Sphere grid. Recent results have shown that the corresponding Euler-Maclaurin expansion can be surprisingly simplified in terms of derivatives evaluated at the eight corners of a cube. This suggests correcting the quadrature weights near the corners to increase the accuracy. The paper is devoted to this problem. Taking benefit from reduced forms of the quadrature error, we derive new corrections up to the tenth-order accuracy. The proposed corrections are slight adjustements of the usual metric of the Cubed Sphere, around the corners. All of the weights have closed formula; they are positive and have the octahedral symmetry. The accuracy of the rules is assessed in numerical experiments, which include a comparison with an interpolatory approach.
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BibTeX
@misc{bellet2026highorder,
title = {{High-order corrections of the trapezoidal rule near the corners of the Cubed Sphere}},
author = {Jean-Baptiste Bellet},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05783859v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - High-order corrections of the trapezoidal rule near the corners of the Cubed Sphere
AU - Jean-Baptiste Bellet
PY - 2026
DA - 2026/10/07
PB - HAL
UR - https://hal.science/hal-05783859v1
AB - The trapezoidal rule allows to define a simple spherical quadrature rule on the Cubed Sphere grid. Recent results have shown that the corresponding Euler-Maclaurin expansion can be surprisingly simplified in terms of derivatives evaluated at the eight corners of a cube. This suggests correcting the quadrature weights near the corners to increase the accuracy. The paper is devoted to this problem. Taking benefit from reduced forms of the quadrature error, we derive new corrections up to the tenth-order accuracy. The proposed corrections are slight adjustements of the usual metric of the Cubed Sphere, around the corners. All of the weights have closed formula; they are positive and have the octahedral symmetry. The accuracy of the rules is assessed in numerical experiments, which include a comparison with an interpolatory approach.
ER -