# New Second Order Up-wind Scheme for Oblique Derivative Boundary Value Problem

## Main Article Content

## Abstract

This work is devoted to solving the Laplace equation with an oblique derivative prescribed as a boundary condition on a non-uniform logically rectangular grids. Laplace equation is solved using a finite volume method and we use new up-wind type discretization for the oblique derivative. In order to approximate Laplace equation on non-uniform 3D meshes,the normal derivative is split into the tangential derivative on finite volume faces and derivative in the direction of the vector connecting representative points of finite volumes. New second order up-wind discretization of the oblique derivative, based on linear reconstruction of solution on 3D grid, is presented. A gradient is used for a better approximation of unknown value on the boundary of finite volume. Since both, up-wind and finite volume method, are second order, the whole scheme is second order.

## Article Details

How to Cite

MEDĽA, Matej; MIKULA, Karol.
New Second Order Up-wind Scheme for Oblique Derivative Boundary Value Problem.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 254-263, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/414>. Date accessed: 22 sep. 2017.
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## References

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[8] Z. Minarechová, M.Macák, R. Čunderlík, K.Mikula, High-resolution global gravity field modelling by the finite volume method, Studia Geophysica et Geodaetica, Vol. 59 (2015) pp. 1-20

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[2] M. Húska, M. Medľa, K. Mikula, P. Novysedlák, M. Remešíková, A new form-finding method based on mean curvature flow of surfaces, ALGORITMY 2012, 19th Conference on Scientific Computing, Podbanske, Slovakia, September 9-14, 2012, Proceedings of contributed papers and posters (Eds. A.Handloviˇ cov ́ a, Z.Min ́ arechov ́ a, D.Sevˇ ˇ coviˇ c), ISBN 978-80-2273742-5, Publishing House of STU, 2012, pp. 120-131

[3] Koch K.R. and Pope A.J., Uniqueness and existence for the geodetic boundary value problem using the known surface of the earth, Bull. Geod., 46, 1972, 467-476

[4] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, 2002, ISBN: 978-0521009249

[5] M. Macák, Numerická metódy v geodézii, PhD thesis, Faculty of Civil Engineering, Slovak University of Technology Bratislava, 2014

[6] M. Macák, K. Mikula, Z. Minarechová, R. Čunderlík, On an iterative approach to solving the nonlinear satellite-fixed geodetic boundary-value problem, International Association of Geodesy Symposia, Proceedings of VIII Hotine Marussi Symposium 2013, Rome, Italy, June 17-21, 2013, accepted

[7] M. Medľa, Tvorba ”optimálnych” logicky štvoruholníkových sietí v 2D a 3D oblastiach nad topografiou Zeme, bachelor thesis, Faculty of Civil Engineering, Slovak University of Technology Bratislava, 2012

[8] Z. Minarechová, M.Macák, R. Čunderlík, K.Mikula, High-resolution global gravity field modelling by the finite volume method, Studia Geophysica et Geodaetica, Vol. 59 (2015) pp. 1-20

[9] G.L.G. Sleijpen, D.R. Fokkema, Bicgstab(l) for Linear Equations Involving Unsymmetric Matrices with Complex Spectrum, 1993, http://dspace.library.uu.nl/handle/1874/16827