# Complex of Models, High-Resolution Schemes and Programs for the Predictive Modeling of Suffocation in Shallow Waters

## Abstract

The paper covers the development and research of mathematical models of the algal bloom, causing suffocations in shallow waters on the basis of modern information technologies and computational methods, by which the accuracy of predictive modeling of the ecology situation of coastal systems is increased. Developed model takes into account the follows: the water transport; microturbulent diffusion; gravitational sedimentation of pollutants and plankton; nonlinear interaction of plankton populations; biogenic, temperature and oxygen regimes; influence of salinity. The computational accuracy is significantly increased and computational time is decreased at using schemes of high order of accuracy for discretization of the model. The practical significance is the software implementation of the proposed model, the limits and prospects of it practical use are defined. Experimental software was developed based on multiprocessor computer system and intended for mathematical modeling of possible progress scenarios of shallow waters ecosystems on the example of the Azov Sea in the case of suffocation. We used decomposition methods of grid domains in parallel implementation for computationally laborious convection-diffusion problems, taking into account the architecture and parameters of multiprocessor computer system. The advantage of the developed software is also the use of hydrodynamical model including the motion equations in the three coordinate directions.

## Keywords

Mathematical model Water bloom Suffocation Phytoplankton Multiprocessor computer system Computational experiments## References

- 1.Samarsky, A.A., Nikolaev, E.S.: Methods of solving grid equations, p. 588. Science, Moscow (1978). (in Russian)Google Scholar
- 2.Sukhinov, A.I., Chistyakov, A.E.: Adaptive modified alternating triangular iterative method for solving grid equations with non-selfadjoint operator. Math. Model.
**24**(1), 3–20 (2012). (in Russian)zbMATHGoogle Scholar - 3.State Research Center Planeta. http://planet.iitp.ru/english/index_eng.htm
- 4.Samarskiy, A.A.: Theory of Difference Schemes. M. Nauka (1989). (in Russian)Google Scholar
- 5.Konovalov, A.N.: The method of steepest descent with adaptive alternately-triangular preamplification. Diff. Eqn.
**40**(7), 953 (2004). (in Russian)Google Scholar - 6.Konovalov, A.N.: The theory of alternating-triangular iterative method. Siberian Math. J.
**43**(3), 552 (2002). (in Russian)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Sukhinov, A.I., Chistyakov, A.E., Shishenya, A.V.: Error estimate of the solution of the diffusion equation on the basis of the schemes with weights. Math. Model.
**25**(11), 53–64 (2013). (in Russian)zbMATHGoogle Scholar - 8.Chetverushkin, B., Gasilov, V., Iakobovski, M., Polyakov, S., Kartasheva, E., Boldarev, A., Abalakin, I., Minkin, A.: Unstructured mesh processing in parallel CFD project GIMM. In: Parallel Computational Fluid Dynamics, pp. 501–508. Elsevier, Amsterdam (2005)Google Scholar
- 9.Petrov, I.B., Favorsky, A.V., Sannikov, A.V., Kvasov, I.E.: Grid-characteristic method using high order interpolation on tetrahedral hierarchical meshes with a multiple time step. Math. Model.
**25**(2), 42–52 (2013). (in Russian)zbMATHGoogle Scholar - 10.Sukhinov, A.I., Chistyakov, A.E., Semenyakina, A.A., Nikitina, A.V.: Parallel realization of the tasks of the transport of substances and recovery of the bottom surface on the basis of high-resolution schemes. Comput. Methods Program. New Comput. Technol.
**16**(2), 256–267 (2015). (in Russian)Google Scholar - 11.Chistyakov, A.E., Hachunts, D.S., Nikitina, A.V., Protsenko, E.A., Kuznetsova, I.: Parallel Library of iterative methods of the SLAE solvers for problem of convection-diffusion-based decomposition in one spatial direction. Modern Prob. Sci. Educ. (1–1), 1786 (2015). (in Russian)Google Scholar
- 12.Sukhinov, A.I., Nikitina, A.V., Semenyakina, A.A., Protsenko, E.A.: Complex programs and algorithms to calculate sediment transport and multi-component suspensions on a multiprocessor computer system. Eng. J. Don
**38**(4(38)), 52 (2015). (in Russian)Google Scholar - 13.Nikitina, A.V., Abramenko, Y.A., Chistyakov, A.E.: Mathematical modeling of oil spill in shallow water bodies. Inf. Comput. Sci. Eng. Educ.
**3**(23), 49–55 (2015). (in Russian)Google Scholar - 14.Chistyakov, A.E., Nikitina, A.V., Ougolnitsky, G.A., Puchkin, V.M., Semenov, I.S., Sukhinov, A.I., Usov, A.B.: A differential game model of preventing fish kills in shallow water bodies. Game Theory Appl.
**17**, 37–48 (2015)zbMATHGoogle Scholar - 15.Sukhinov, A.I., Nikitina, A.V., Semenyakina, A.A., Chistyakov, A.E.: A set of models, explicit regularized high-resolution schemes and programs for predictive modeling of consequences of emergency oil spill. In: Proceedings of the International Scientific Conference on Parallel Computational Technologies (PCT 2016), pp. 308–319 (2016). (in Russian)Google Scholar
- 16.Nikitina, A.V., Semenyakina, A.A., Chistyakov, A.E.: Parallel implementation of the tasks of diffusion-convection-based high-resolution schemes. Vestnik Comput. Inf. Technol.
**7**(145), 3–8 (2016). (in Russian)Google Scholar - 17.Sukhinov, A.I., Chistyakov, A.E., Semenyakina, A.A., Nikitina, A.V.: Numerical modeling of an ecological condition of the Azov Sea with application of high-resolution schemes on the multiprocessor computing system. Comput. Res. Model.
**8**(1), 151–168 (2016). (in Russian)Google Scholar - 18.Sukhinov, A.I., Nikitina, A.V., Semenyakina, A.A., Chistyakov, A.E.: Complex of models, explicit regularized high-resolution schemes and applications for predictive modeling of after-math of emergency oil spill. In: CEUR Workshop Proceedings, vol. 1576, pp. 308–319, 10th Annual International Scientific Conference on Parallel Computing Technologies, PCT 2016; Arkhangelsk; Russian Federation; 29 March 2016 through 31 March 2016; Code 121197 (2016). (in Russian)Google Scholar
- 19.Sukhinov, A.I., Nikitina, A.V., Chistyakov, A.E., Semenov, I.S., Semenyakina, A.A., Khachunts, D.S.: Mathematical modeling of eutrophication processes in shallow waters on multiprocessor computer system. In: CEUR Workshop Proceedings, vol. 1576, pp. 320–333, 10th Annual International Scientific Conference on Parallel Computing Technologies, PCT 2016; Arkhangelsk; Russian Federation; 29 March 2016 through 31 March 2016; Code 121197 (2016). (in Russian)Google Scholar
- 20.Nikitina, A.V., Sukhinov, A.I., Ougolnitsky, G.A., Usov, A.B., Chistyakov, A.E., Puchkin, M.V., Semenov, I.S.: Optimal management of sustainable development at the biological rehabilitation of the Azov Sea. Math. Model.
**28**(7), 96–106 (2016). (in Russian)zbMATHGoogle Scholar