On an Iterative Approach to Solving the Nonlinear Satellite-Fixed Geodetic Boundary-Value Problem

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not assumed to have a specific form, but only to satisfy some (usual) coercivity and consistency properties. The oblique boundary conditions are split into a normal component, which directly appears in the flux balance on control volumes touching the domain boundary, and a tangential component which is managed as an advection term on the boundary. This advection term is discretised using a finite volume method based on a centred discretisation (to ensure optimal rates of convergence) and stabilised using a vanishing boundary viscosity. A convergence analysis, based on the 3rd Strang Lemma [9], is conducted in this generic finite volume framework, and yields the expected O(h) optimal convergence rate in discrete energy norm.We then describe a specific choice of numerical fluxes, based on a generalised hexahedral meshing of the computational domain. These fluxes are a corrected version of fluxes originally introduced in [29]. We identify mesh regularity parameters that ensure that these fluxes satisfy the required coercivity and consistency properties. The theoretical rates of convergence are illustrated by an extensive set of 3D numerical tests, including some conducted with two variants of the proposed scheme. A test involving real-world data measuring the disturbing potential in Earth gravity modelling over Slovakia is also presented.

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“…See Figure 1 for an illustration. Such nonlinear satellite-fixed geodetic BVP [28] for the disturbing potential T can be formulated as follows…”

Section: G(x) = U (X) + T (X)mentioning

confidence: 99%

“…These schemes are based either on upwind or central approximation of the oblique derivative. A numerical approximation of a nonlinear problem with eikonal-type boundary condition (4) was presented in [28]. This approximation is based on an iterative update of the oblique derivative condition.…”

Section: Introductionmentioning

confidence: 99%

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Droniou

Medl’a

Mikula

2019

Journal of Computational Physics

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“…Similarly, geodetic researchers appeal to numerical schemes for solving the CBVPs, in view of the high-degree nonlinearity of gravitational fields with complex mass distributions. However, this has not been realized up to now, despite various attempts applying various numerical methods, e.g., the finite element method (FEM;Meissl 1981;Bian and Chao 1991;Fašková et al 2010), the boundary element method (BEM; Klees 1995;Lehmann and Klees 1999;Čunderlík et al 2008) and the finite volume method (FVM; Macák et al 2012Macák et al , 2015Medl'a et al 2018).…”

Section: Introductionmentioning

confidence: 99%

Modeling the gravitational field by using CFD techniques

Yin¹,

Sneeuw

2021

J Geod

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A harmonic scalar field has a Laplacian (i.e., both source-free and curl-free) gradient vector field and vice versa. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, which is known as the complex-boundary value problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a Laplacian vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of the comet 67P/Churyumov–Gerasimenko which has an irregular shape. The method is validated in a closed-loop simulation by comparing the result with a direct integration of Newton’s formula. Both methods are consistent with a relative magnitude discrepancy at the percentage level and with a small directional difference root-mean-square value of $$0.78^{\circ }$$ 0 . 78 ∘ . Moreover, the Laplacian property of the potential flow’s velocity field is proved mathematically. From both theoretical and practical points of view, the new numerical method is able to overcome the divergence problem and, hence, has a good potential for solving CBVPs.

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On an Iterative Approach to Solving the Nonlinear Satellite-Fixed Geodetic Boundary-Value Problem

Cited by 2 publications

References 13 publications

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Design and analysis of finite volume methods for elliptic equations with oblique derivatives; application to Earth gravity field modelling

Modeling the gravitational field by using CFD techniques

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