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Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography

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Abstract

The paper presents a numerical solution of the oblique derivative boundary value problem on and above the Earth’s topography using the finite volume method (FVM). It introduces a novel method for constructing non-uniform hexahedron 3D grids above the Earth’s surface. It is based on an evolution of a surface, which approximates the Earth’s topography, by mean curvature. To obtain optimal shapes of non-uniform 3D grid, the proposed evolution is accompanied by a tangential redistribution of grid nodes. Afterwards, the Laplace equation is discretized using FVM developed for such a non-uniform grid. The oblique derivative boundary condition is treated as a stationary advection equation, and we derive a new upwind type discretization suitable for non-uniform 3D grids. The discretization of the Laplace equation together with the discretization of the oblique derivative boundary condition leads to a linear system of equations. The solution of this system gives the disturbing potential in the whole computational domain including the Earth’s surface. Numerical experiments aim to show properties and demonstrate efficiency of the developed FVM approach. The first experiments study an experimental order of convergence of the method. Then, a reconstruction of the harmonic function on the Earth’s topography, which is generated from the EGM2008 or EIGEN-6C4 global geopotential model, is presented. The obtained FVM solutions show that refining of the computational grid leads to more precise results. The last experiment deals with local gravity field modelling in Slovakia using terrestrial gravity data. The GNSS-levelling test shows accuracy of the obtained local quasigeoid model.

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Acknowledgements

This article is also based upon work from COST Action IC1406 High-Performance Modelling and Simulation for Big Data Applications (cHiPSet), supported by COST (European Cooper-ation in Science and Technology).

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Authors and Affiliations

  1. Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05, Bratislava, Slovakia

    Matej Medl’a, Karol Mikula, Róbert Čunderlík & Marek Macák

  2. Algoritmy:SK s.r.o., Sulekova 6, 81106, Bratislava, Slovakia

    Karol Mikula

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  1. Matej Medl’a
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  2. Karol Mikula
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  3. Róbert Čunderlík
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  4. Marek Macák
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Correspondence to Matej Medl’a.

Additional information

This work was supported by the Grants APVV-15-0522, VEGA 1/0608/15 and VEGA 1/0714/15 and STU Grant scheme for Support of Young Researchers.

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Medl’a, M., Mikula, K., Čunderlík, R. et al. Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography. J Geod 92, 1–19 (2018). https://doi.org/10.1007/s00190-017-1040-z

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  • DOI: https://doi.org/10.1007/s00190-017-1040-z

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