About correctness of the suspension transport and sedimentation model, taking into account bottom relief changes
https://doi.org/10.23947/2587-8999-2018-2-76-90
Abstract
This paper is devoted to the study of the spatial-three-dimensional model of sedimentation of suspended particles in the coastal zone, taking into account changes in the bottom topography. The model takes into account the following processes: advective transfer due to movement of the aquatic environment, microturbulent diffusion and gravitational sedimentation of suspended particles, as well as changes in bottom geometry caused by sedimentation of suspended particles or rise of sediment particles. A change in the bottom relief leads to the need to solve an initial-boundary value problem for an equation of parabolic type with lower derivatives in a domain whose geometry depends on the desired solution function, which leads, in general, to a non-linear formulation of the problem. The model was linearized on a time grid due to the «freezing» of the bottom relief within one time step and the subsequent recalculation of the bottom surface function based on the changed function of suspended matter concentration, as well as a possible change in the velocity vector of the aquatic environment. For the linearized problem, a quadratic functional was constructed and the energy method proved the uniqueness of the solution of the corresponding initial-boundary problem within an arbitrary time step. Based on the transformation of the quadratic functional, an a priori estimate of the solution norm in the L2 functional space was obtained depending on the integral estimates for the right side and the initial condition, and, thus, the stability of the solution of the initial problem against the change of the initial and boundary conditions and functions of the right side. The model may be of value in predicting the spread of pollution and changes in the bottom topography, both under anthropogenic impact and due to naturally occurring natural processes in the coastal zone.
Keywords
coastal systems,
mathematical model,
diffusion-convection problems of deposition of suspended matter,
change of the bottom relief,
uniqueness of the solution and stability of the initial-boundary value problem
Supporting agencies: The research is done on theme no. 6905.2017/БЧ within the frame of the government task of the Ministry of Education and Science of the Russian Federation in R&D.
About the Authors
Alexander Ivanovich Sukhinov
Don State Technical University
(1st Gagarin Square, Rostov-on-Don, Russian Federation)
Russian Federation
Russian Federation
Sukhinov Alexander Ivanovich, Don State Technical University (1st Gagarin Square, Rostov-on-Don, Russian Federation), Doctor of Science in Physics and Maths, Professor
Valentina Vladimirovna Sidoryakina
Taganrog Institute of A.P. Chekhov (branch) RSUE
(Initiative Street, Taganrog, Russian Federation)
Russian Federation
Russian Federation
Sidoryakina Valentina Vladimirovna, Taganrog Institute of A.P. Chekhov (branch) RSUE (Initiative Street, Taganrog, Russian Federation), Candidate of Science in Physics and Maths, Associate professor
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Sukhinov A.I., Sidoryakina V.V. About correctness of the suspension transport and sedimentation model, taking into account bottom relief changes. Computational Mathematics and Information Technologies. 2018;2(2). https://doi.org/10.23947/2587-8999-2018-2-76-90
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