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

• Published in: Mathematics (Basel) Vol. 11; no. 3; p. 575
• Main Authors: Sukhinov, Alexander, Belova, Yulia, Panasenko, Natalia, Sidoryakina, Valentina
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2023
• Subjects: Biochemistry; biogeochemical cycles; Biogeochemistry; Biological activity; Biological models (mathematics); Boundary value problems; Chains; Chemical elements; Climate change; Coastal processes; Coasts; Convergence; Ecosystems; energy method; Energy methods; Fluid mechanics; Geochemical cycles; Hilbert space; Hydrology; Linearization; mathematical model; Mathematical models; Nitrogen; Nonlinear theories; Numerical analysis; Numerical methods; Phosphorus; Plankton; Salinity; solutions proximity; Spatial distribution; Three dimensional models; uniqueness of solution; Russia; Azov Sea
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11030575

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 L2 as the time step τ tends to zero at the rate O(τ).