A novel scheme for solving the oblique derivative boundary-value problem

Abstract

The paper is devoted to a novel scheme for solving the boundary-value problem (BVP) with the oblique derivative boundary condition (BC). In this approach, the oblique derivative in the BC is decomposed into its normal and two tangential components which are approximated by means of numerical solution values. Then the numerical scheme by the finite volume method is developed and testing numerical experiments are done. The obtained numerical solutions are compared to the exact one to show that the proposed method is second order accurate. Afterwards, the algorithm is applied to solving the fixed gravimetric BVP, namely, the numerical solution is sought in a domain bounded by a part of the Earth’s surface (Himalaya region or Slovakia), four side boundaries and a corresponding upper boundary at the satellite level. On the Earth’s surface, the oblique derivative BC in the form of surface gravity disturbances from the EGM2008, DTU10-GRAV or the detailed gravity mapping is taken into account. On the upper and side boundaries, the Dirichlet BC from the EGM2008 or GOCO03s is applied. The disturbing potential as a direct numerical result is compared with the solution to the more common BVP with the Neumann BC considered on the Earth’s surface. All numerical experiments show better agreement of the solution to the BVP with the oblique derivative BC than solution to the BVP with the Neumann BC in comparison with the disturbing potential obtained by a different mathematical approach. In area of Slovakia, when applying the GPS/levelling test at 61 points, we have gained 1.7 cm improvement in favour of the standard deviation of residuals of quasigeoidal heights obtained by solving the BVP with the oblique derivative BC in comparison with the solution to BVP with the Neumann BC.