Geodesic
A semi-splitting finite difference scheme of matrix coefficients and its application to describe the tsunami propagation
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 2, pp. 44-51 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The well-known finite-difference scheme of Moretti of splitting the matrix coefficients of the system of gas dynamics equations involves writing equations in a special form — pressure and internal energy are excluded from the equation using the equations of state for an ideal gas. In this paper the author proposes a modification of Moretti scheme as a finite-difference scheme of semi-splitting of matrix coefficients which do not intend to constitute a system of equations in a special form. The semi-splitting scheme allows solving equations of hyperbolic equations of state of any type, for example, even those in tabular form. For one-dimensional equations of propagation of circular wave on the water surface, that are the equations of hyperbolic type, the results of the calculations of the problem of the propagation of a surface wave in the ocean and the output of wave on the shore of the ocean area are given according to one-dimensional computational code of the shallow water theory. Verification of semi-splitting finite-difference scheme is performed by comparing the calculation results for the problem of the propagation of a single surface wave in the ocean and the problem of propagation of a wave train on the ocean surface with the results of calculations of the same problems cited in the work by C. Mader. To calculate the wave setup on the shore the approach is used in the computational code of the shallow water theory, which is described in the work by An. G. Marchuk, A. A. Anisimov. By comparing the calculation results with analytical solutions, the suitability of the computational code proposed in this work for the problem solution of the sea wave setup on land is demonstrated. Further development work is seen in the development of a two-dimensional program for calculating the surface wave propagation.
Keywords: semi-splitting finite-difference scheme, shallow water theory, seashore, astronomical body.
Mots-clés : tsunami 
@article{VYURM_2016_8_2_a5,
     author = {V. A. Simonenko and N. A. Skorkin and A. S. Uglov},
     title = {A semi-splitting finite difference scheme of matrix coefficients and its application to describe the tsunami propagation},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {44--51},
     year = {2016},
     volume = {8},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2016_8_2_a5/}
}
TY  - JOUR
AU  - V. A. Simonenko
AU  - N. A. Skorkin
AU  - A. S. Uglov
TI  - A semi-splitting finite difference scheme of matrix coefficients and its application to describe the tsunami propagation
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2016
SP  - 44
EP  - 51
VL  - 8
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VYURM_2016_8_2_a5/
LA  - ru
ID  - VYURM_2016_8_2_a5
ER  - 
%0 Journal Article
%A V. A. Simonenko
%A N. A. Skorkin
%A A. S. Uglov
%T A semi-splitting finite difference scheme of matrix coefficients and its application to describe the tsunami propagation
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2016
%P 44-51
%V 8
%N 2
%U http://geodesic.mathdoc.fr/item/VYURM_2016_8_2_a5/
%G ru
%F VYURM_2016_8_2_a5
V. A. Simonenko; N. A. Skorkin; A. S. Uglov. A semi-splitting finite difference scheme of matrix coefficients and its application to describe the tsunami propagation. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 2, pp. 44-51. http://geodesic.mathdoc.fr/item/VYURM_2016_8_2_a5/

[1] Fletcher C. A. J., Computational Techniques for Fluid Dynamics, v. 2, Springer Verlag Ed., 1988 | Zbl

[2] Kuropatenko V. F., “Methods of shock wave calculations”, Dal'nevostochnyy matematicheskiy zhurnal, 2:2 (2001), 45–59 | MR

[3] L. Charles Mader, “Numerical Simulation of Tsunami”, Journal of Physical Oceanography, 4 (1974), 74–82 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[4] A. G. A. Marchuk, A. A. Anisimov, “Method for Numerical Modeling of Tsunami Run-up on the Coast of Arbitrary Profile”, ITS 2001 Processing, Session 7, Number 7-27, 933–940; Препринт No 25, ИПФ АН СССР, Горький, 1981, 15 с.

[5] Mazova R. Kh., Pelinovskiy E. N., “Linear Theory of Wave Run Up on a Beach”, Izv. AN SSSR. Fizika atmosfery i okeana, 18:2 (1982), 166–171 (in Russ.) | MR | Zbl

[6] Simonenko V. A., Skorkin N. A., Elsukov V. P., Uglov A. S., “Mathematical Modeling of Magatsunamis”, Vestnik YuUrGU, ser. “Matematika, fizika, khimiya”, 2008, no. 22(11), 62–69 (in Russ.) | MR

[7] Beizel S. A., Shokina N. Y., Khakimzyanov G. S., Chubarov L. B., Kovyrkina O. A., Ostapenko V. V., “On some numerical algorithms for computation of tsunami runup in the framework of shallow water model, I”, Computational technologies, 19:1 (2014), 40–62 (in Russ.)

[8] M. J. Ketabdaria, M. R. H. Nobarib, M. Moradi Larmaei, “Simulation of waves group propagation and breaking in coastal zone using a Navier-Stokes solver with an improved VOF free surface treatment”, Appl. Ocean Res., 30:2 (2008), 130–143 | DOI

[9] Kochin N. E., Kibel' I. A., Roze N. V., Theoretical Hydromechanics Theoretical Hydromechanics, v. II, Fizmatlit Publ., M., 1963, 728 pp. (in Russ.)