On the Class of Compact Grid-Characteristic Schemes

• Nikolay I. Khokhlov
• Vasily I. Golubev
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)

Abstract

This work is devoted to the numerical solution of the linear hyperbolic transport equation. The importance of this topic is explained by the wide variety of possible applications: seismic survey of oil and gas deposits, seismic resistance of ground facilities estimation, hydrodynamic simulation. In one-dimensional case, a class of two-point compact difference schemes is considered. With the usage of appropriate procedure for polynomial interpolation and analysis of obtained solution behavior second-third order of approximation is achieved for continuous case. Proposed schemes also produce fewer oscillations on discontinuous solutions. This approach was generalized to the two-dimensional case. The technique of splitting in coordinates and parallelepiped meshes were successfully used. The influence of separate steps of the computational algorithm on the accuracy of the resulting scheme is investigated. The best results were obtained by storing orthogonal (additional) derivatives; however, it increased the random access memory consumption. With the same basic principle compact grid-characteristic schemes for the three-dimensional case were constructed. Due to the increase of the computational complexity of the algorithm the parallelization with Massively Parallel Processing (MPI) technology was done. To keep the order of approximation two more orthogonal derivatives must be stored. A set of computational simulation was carried out in this work for comparison with analytical solutions.

Keywords

Computer simulation Compact schemes Grid-Characteristic method Transport equation

Notes

Acknowledgements

This work was supported by the Russian Foundation for Basic Research, project no. 16-08-00212_a.

References

1. 1.
Belotserkovskii, O.M.: Numerical Modeling on the Mechanics of Continuous Media. Fizmatlit, Moscow (1994). (in Russian)Google Scholar
2. 2.
Rusanov, V.: Calculation of intersection of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)Google Scholar
3. 3.
Grudnitskii, V.G., Prokhorchuk, YuA: A method for constructing difference schemes of any order of accuracy for partial differential equations. Dokl. Akad. Nauk SSSR 6, 1249–1252 (1977). (in Russian)Google Scholar
4. 4.
Stupitsky, E.L., Kholodov, A.S., Repin, A.Y., Kholodov Ya.A.: Numerical modelling of the behavior of high-energy plasmoid in the upper ionosphere. Comput. Phys. Commun. 164(1–3), 258–261 (2004)Google Scholar
5. 5.
Kholodov, A. S., Kholodov, Ya. A.: Monotonicity criteria for difference schemes designed for hyperbolic equations. Comput. Math. Math. Phys. 46(9), 1560–1588 (2006)
6. 6.
Shokin, YuI, Yanenko, N.N.: Method of Differential Approximation: Application to Gas Dynamics. Nauka, Novosibirsk (1985). (in Russian)
7. 7.
Tolstykh, A.I.: Compact difference schemes and their application to aero hydrodynamics problems. Nauka, Moscow (1990). (in Russian)Google Scholar
8. 8.
Rogov, B.V., Mikhailovskaya, M.N.: Monotonic bi-compact schemes for linear transport equations. Math. Models Comput. Simul. 4(1), 92–100 (2012)
9. 9.
Yabe, T., Aoki, T., Sakaguchi, G., Wang, P.Y., Ishikawa, T.: The compact CIP (Cubic-Interpolated Pseudo-particle) method as a general hyperbolic solver. Comput. Fluids 19(3–4), 421–431 (1991)
10. 10.
Yabe, T., Mizoe, H., Takizawa, K., Moriki, H., Im, H.-N., Ogata, Y.: Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme. J. Comput. Phys. 194(1), 57–77 (2004)
11. 11.
Petrov, I.B., Khokhlov, N.I.: Compact grid-characteristical scheme for linear transfer equation. Modeling of Information Processing. Collection of Articles (Mosk. Fiz. Tekh. Inst., Moscow) 18–22 (2014). (in Russian)Google Scholar
12. 12.
Magomedov, K.M., Kholodov, A.S.: The construction of difference schemes for hyperbolic equations based on characteristic relations. USSR Comput. Math. Math. Phys. 9(2), 158–176 (1969)
13. 13.
Voinov, O.Ya., Golubev, V.I., Petrov, I.B.: Elastic imaging using multiprocessor computer systems. In: CEUR Workshop Proceedings, vol. 1787, pp. 491–495 (2016)Google Scholar
14. 14.
Golubev, V.I., Gilyazutdinov, R.I., Petrov, I.B., Khokhlov, N.I., Vasyukov, A.V.: Simulation of dynamic processes in three-dimensional layered fractured media with the use of the grid-characteristic numerical method. J. Appl. Mech. Tech. Phys. 58(3), 539–545 (2018)
15. 15.
Golubev, V.I., Voinov, O.Y., Zhuravlev, Y.I.: On seismic imaging of fractured geological media. Dokl. Math. 96(2), 514–516 (2017)
16. 16.
Beklemysheva, K.A., Vasyukov, A.V., Golubev, V.I., Zhuravlev, YuI: On the estimation of seismic resistance of modern composite oil pipeline elements. Dokl. Math. 97(2), 184–187 (2018)
17. 17.
Golubev, V.I., Petrov, I.B., Khokhlov, N.I.: Compact grid-characteristic schemes of higher orders of accuracy for a 3D linear transport equation. Math. Models Comput. Simul. 8(5), 577–584 (2016)