Research of the Solutions Proximity of Linearized and Nonlinear Problems of the Biogeochemical Process Dynamics in Coastal Systems

Alexander Sukhinov, Yulia Belova (), Natalia Panasenko and Valentina Sidoryakina
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Alexander Sukhinov: Department of Mathematics and Computer Science, Don State Technical University, 344000 Rostov-on-Don, Russia
Yulia Belova: Department of Mathematics and Computer Science, Don State Technical University, 344000 Rostov-on-Don, Russia
Natalia Panasenko: Department of Mathematics and Computer Science, Don State Technical University, 344000 Rostov-on-Don, Russia
Valentina Sidoryakina: Department of Mathematics and Computer Science, Don State Technical University, 344000 Rostov-on-Don, Russia

Mathematics, 2023, vol. 11, issue 3, 1-15

Abstract: The article considers a non-stationary three-dimensional spatial mathematical model of biological kinetics and geochemical processes with nonlinear coefficients and source functions. Often, the step of analytical study in models of this kind is skipped. The purpose of this work is to fill this gap, which will allow for the application of numerical modeling methods to a model of biogeochemical cycles and a computational experiment that adequately reflects reality. For this model, an initial-boundary value problem is posed and its linearization is carried out; for all the desired functions, their final spatial distributions for the previous time step are used. As a result, a chain of initial-boundary value problems is obtained, connected by initial–final data at each step of the time grid. To obtain inequalities that guarantee the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems, the energy method, Gauss’s theorem, Green’s formula, and Poincaré’s inequality are used. The scientific novelty of this work lies in the proof of the convergence of solutions of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space L 2 as the time step τ tends to zero at the rate O(τ).

Keywords: mathematical model; biogeochemical cycles; linearization; uniqueness of solution; solutions proximity; energy method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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