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A model of transport and transformation of biogenic elements in the coastal system and its numerical implementation. (English. Russian original) Zbl 1404.86002
Comput. Math. Math. Phys. 58, No. 8, 1316-1333 (2018); translation from Zh. Vychisl. Mat. Mat. Fiz. 58, No. 8 (2018).
Summary: A discrete mathematical model of hydrobiology of coastal zone is constructed and analyzed. The model takes into account the transport and transformation of polluting biogenic elements in water basins. The propagation and transformation of biogenic elements is affected by such physical factors as three-dimensional motion of water taking into account the advective transport and microturbulent diffusion, spatially inhomogeneous distribution of temperature, salinity, and oxygen. Biogenic pollutants typically arrive into the water basin with river flow, which depends on the weather and climate of the geographic region, or with drainage of insufficiently purified domestic and industrial waste or other kinds of anthropogenic impact. Biogenic pollutants can also appear due to secondary pollution processes, such as stirring up and transport of bed silt, shore abrasion, etc. Stoichiometric relations between biogenic nutrients for phytoplankton algae that can be used to determine the limiting nutrient for each species are obtained. Observation models describing the consumption, accumulation of nutrients, and growth of phytoplankton are considered. A three-dimensional mathematical model of transformation of forms of phosphorus, nitrogen, and silicon in the plankton dynamics problem in coastal systems is constructed and analyzed. This model takes into account the convective and diffusive transport, absorption, and release of nutrients by phytoplankton as well as transformation cycles of phosphorus, nitrogen, and silicon forms. Numerical methods for solving the problem that are based on high-order weighted finite difference schemes and take into account the degree of fill of the computation domain control cells are developed. These methods are implemented on a multiprocessing system. They make it possible to improve the accuracy of the numerical solution and decrease the computation time by several fold. Based on the numerical implementation, dangerous phenomena in coastal systems related to the propagation of pollutants, including oil spill, eutrophication, and algae bloom, which causes suffocation phenomena in water basins, are reconstructed.
MSC:
86-08 Computational methods for problems pertaining to geophysics
92D40 Ecology
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
References:
[1] G. I. Marchuk, Mathematical Modeling of the Environment (Nauka, Moscow, 1982) [in Russian]. · Zbl 0493.90001
[2] Davydov, A. A.; Chetverushkin, B. N.; Shil’nikov, E. V., Simulating flows of incompressible and weakly compressible fluids on multicore hybrid computer systems, Comput. Math, Math. Phys., 50, 2157-2165, (2010) · Zbl 1224.65326
[3] Chetverushkin, B. N., Resolution limits of continuous media mode and their mathematical formulations, Math. Models Comput. Simul., 5, 266-279, (2013)
[4] A. I. Sukhinov and A. A. Sukhinov, “Reconstruction of 2001 ecological disaster in the Azov Sea on the basis of precise hydrophysics models,” in Proc. of CFD 2004 Conference on Parallel Computational Fluid Dynamics, Multidisciplinary Applications, Las Palmas de Gran Canaria, Spain (Elsevier, Amsterdam, 2005), pp. 231-238.
[5] Sukhinov, A. I.; Chistyakov, A. E.; Protsenko, E. A., Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs, Math. Models Comput. Simul., 6, 351-363, (2014) · Zbl 1357.65135
[6] A. Semenyakina and S. Protsenko, “Complex of parallel programs for modeling oil products transport in coastal systems,” in MATEC Web of Conferences, 2017. 132, 04016 (2017). doi 10.1051/matecconf/201713204016
[7] Steele, J. H.; Henderson, E. W., A simple model for plankton patchiness, J. Plankton Research, 14, 1397-1403, (1992)
[8] Gushchin, V. A.; Matyushin, P. V., Numerical simulation and visualization of vortical structure transformation in the flow past a sphere at an increasing degree of stratification, Comput. Math. Math. Phys., 51, 251-263, (2011) · Zbl 1224.76074
[9] Sukhinov, A. I.; Chistyakov, A. E., A parallel implementation of the three-dimensional model of shallow reservoir hydrodynamics on a supercomputer, Vychisl. Metody Program. Nov. Vychisl. Tekhnol., 13, 290-297, (2012)
[10] Yakushev, E. V.; Lukashev, Yu. F.; Skirta, A. Yu.; Sorokin, P. Yu.; Soldatova, E. V.; Yakubenko, V. G.; Sukhinov, A. I.; Sergeev, N. E.; Fomin, S. Yu.; Sapozhnikov, F. V., Comprehensive oceanological studies of the Sea of Azov during cruise 28 of r/v Akvanavt (July-August 2001), Oceanology, 43, 39-47, (2003)
[11] Alekseenko, E.; Roux, B.; Sukhinov, A.; Kotarba, R.; Fougere, D., Coastal hydrodynamics in a windy lagoon, Nonlinear Proc. Geophys., 20, 189-198, (2013)
[12] O. M. Belotserkovskii, Turbulence: New Approaches (Nauka, Moscow, 2003).
[13] V. A. Gushchin and P. V. Matyushin, “Classification of modes of separating fluid flows near a sphere under moderate Reynolds numbers,” in Mathematical Modeling (Nauka, Moscow, 2003), pp. 199-235 [in Russian]. · Zbl 1426.76470
[14] Gushchin, V. A.; Kostomarov, A. V.; Matyushin, P. V., 3D Visualization of the separated fluid flows, J. Visualization, 7, 143-150, (2004)
[15] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems (Editorial URSS, Moscow, 1999) [in Russian].
[16] Belotserkovskii, O. M.; Gushchin, V. A.; Shchennikov, V. V., Splitting method as applied to dynamics of viscous incompressible fluid, Zh. Vychisl. Mat. Mat. Fiz., 15, 197-207, (1975)
[17] Petrov, I. B.; Favorskaya, 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. Models Comput. Simul., 5, 409-415, (2013) · Zbl 06686958
[18] A. Sukhinov, A. Nikitina, A. Semenyakina, and A. Chistyakov, “Complex of models, explicit regularized schemes of high-order of accuracy and applications for predictive modeling of after-math of emergency oil spill,” in Proc. of the 10th Annual International Scientific Conference on Parallel Computing Technologies, 2016, Vol. 1576, pp. 308-319.
[19] Nikitina, A. V.; Tret’yakova, M. V., “Simulation of the process of population of a shallow reservoir by Chlorella vulgaris bin,” Izv. Yuzhn. Fed. Univ., Tekhn, Nauki, No., 1, 128-133, (2012)
[20] Yakushev, E. V.; Lukashev, Yu. F.; Skirta, A. Yu.; Sorokin, P. Yu.; Soldatova, E. V.; Yakubenko, V. G.; Sukhinov, A. I.; Sergeev, N. E.; Fomin, S. Yu.; Sapozhnikov, F. V., Comprehensive oceanological studies of the Sea of Azov during cruise 28 of r/v Akvanavt (July-August 2001), Oceanology, 43, 39-47, (2003)
[21] Logofet, D. O., Stronger-than-Lyapunov notions of matrix stability, or how ”flowers” help solve problems in mathematical ecology, Linear Alg. Appl., 398, 75-100, (2005) · Zbl 1062.92072
[22] Tyutyunov, Yu. V.; Titova, L. I.; Senina, I. N., Prey-taxis destabilizes homogeneous stationary state in spatial Gause-Kolmogorov-type model for predator-prey system, Ecol. Complexity , 31, 170-180, (2017)
[23] A. A. Samarskii, Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
[24] A. E. Chistyakov, D. S. Khachunts, A. V. Nikitina, E. A. Protsenko, I. Yu. Kuznetsova, “Library of parallel iterative SLAE solvers for convection-diffusion problem based on the decomposition with respect to one spatial direction,” in Sovrem. Problemy Nauki Obraz., 2015, No. 1-1, p. 1786. http://www.science-education.ru/121-19510
[25] Sukhinov, A. I.; Chistyakov, A. E.; Semenyakina, A. A.; Nikitina, A. V., Numerical simulation of the ecological state of the Sea of Azov using high-order schemes on a multiprocessor computer system, Komput. Issled. Modelir., 8, 1510-168, (2016)
[26] Sukhinov, A. I.; Chistyakov, A. E.; Shishenya, A. V., Error estimate for diffusion equations solved by schemes with weights, Math. Models Comput. Simul., 6, 324-331, (2014) · Zbl 1357.65136
[27] A. V. Nikitina, A. A. Semenyakina, A. E. Chistyakov, E. A. Protsenko, and I. V. Yakovenko, “Application of high-order schemes for solving biological kinetics problems on a multiprocessor computer system,” Fundam. Issled., No. 12-3, 500-504 (2015). http://www.fundamental-research.ru/ru/article/view?id=39569
[28] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).
[29] Konovalov, A. N., The Steepest Descent Method with an Adaptive Alternating-Triangular Preconditioner, Differ. Equations, 40, 1018-1028, (2004) · Zbl 1079.65029
[30] Konovalov, A. N., On the theory of alternate-triangular iterative method, Sib. Mat. Zh., 43, 552-572, (2002) · Zbl 1016.65009
[31] Sukhinov, A. I.; Chistyakov, A. E., Adaptive modified alternate-triangular iterative method for solving grid equations with a non-self-adjoint operator, Math. Models Comput. Simul., 4, 398-409, (2012)
[32] Sukhinov, A. I.; Shishenya, A. V., Increasing Efficiency of Alternating Triangular Method Based on Improved Spectral Estimates, Math. Models Comput. Simul., 5, 257-265, (2013)
[33] A. I. Sukhinov, “Modified alternate-triangular method for heat conduction and filtering problems,” in Computer systems and Algorithms, (Izd-vo Rostov Gos. Univ, Rostov-on-Don, 1984), pp. 52-59 [in Russian]. · Zbl 0592.65072
[34] Adam Coates and Andrew Y. Ng., Learning Feature Representations with K-means (Stanford University, Stanford, 2012).
[35] Analytical GIS Online. http://geo.iitp.ru/index.php
[36] Research Center of Space Hydrometeorology “Planeta”. http://planet.iitp.ru/english/index_eng.htm
[37] A. V. Nikitina, M. V. Puchkin, I. S. Semenov, A. I. Sukhinov, G. A. Ugol’nitskii, A. B. Usov, and A. E. Chistyakov, “A differential-game model of preventing fish suffocation in shallow reservoirs,” in Control of Large-Scale Systems, No. 55 (Institut Problem Upravleniya, Ross. Akad. Nauk, Moscow, 2015), pp. 343-361 [in Russian].
[38] Nikitina, A. V.; Sukhinov, A. I.; Ugol’nitskii, G. A.; Usov, A. B.; Chistyakov, A. E.; Puchkin, M. V.; Semenov, I. S., “Optimal control of sustainable development in the biological rehabilitation of the Azov sea, Math. Models Comput. Simul., 9, 101-107, (2016) · Zbl 1363.86002
[39] Sukhinov, A. I.; Chistyakov, A. E.; Fomenko, N. A., “A procedure for construct finite difference schemes for convection-diffusion-reaction problem taking into account the degree of fill-in of control cells,” Izv. Yuzhn. Fed. Univ., Tekhn, Nauki, No., 4, 87-96, (2013)
[40] A. I. Sukhinov, I. I. Levin, A. E. Chistyakov, A. V. Nikitina, I. S. Semenov, and A. A. Semenyakina, “Solution of the problem of biological rehabilitation of shallow waters on multiprocessor computer system,” in Proc. of the 5th International Conference on Informatics, Electronics and Vision (ICIEV), Dhaka, Bangladesh, 2016, pp. 1128-1133.
[41] Lin, Q.; Lindberg, W. R.; Boyer, D. L.; Fernando, H. J. S., Stratified flow past a sphere, J. Fluid Mech., 240, 315-354, (1992)
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