Accurate computation of gravitational curvature of a tesseroid

In recent years, the fundamental quantity of the gravitational field has been extended from gravitational potential, gravitational vector, and gravitational gradient tensor to gravitational curvature with its first measurement along the vertical direction in laboratory conditions. Previous studies numerically identified the near-zone and polar-region problems for gravitational curvature of a tesseroid, but these issues remain unresolved. In this contribution, we derive the new third-order central and single-sided difference formulas with one, two, and three arguments using the finite difference method. To solve these near-zone and polar-region problems, we apply a numerical approach combining the conditional split, finite difference, and double exponential rule based on these newly derived third-order difference formulas when the computation point is located below, inside, and outside the tesseroid. Numerical experiments with a spherical shell discretized into tesseroids reveal that the accuracy of gravitational curvature is about 4–8 digits in double precision. Numerical results confirm that when the computation point moves to the surface of the tesseroid, the relative and absolute errors of gravitational curvature do not change much, i.e., the near-zone problem can be adequately solved using the numerical approach in this study. When the latitude of the computation point increases, the relative and absolute errors of gravitational curvature do not increase, which solves the polar-region problem with this stable numerical approach. The provided Fortran codes at https://github.com/xiaoledeng/xtessgc-xqtessgc will help with potential applications for the gravitational field of different celestial bodies in geodesy, geophysics, and planetary sciences.