1. Introduction
Every year, the ecological state of the coastal systems of the South of Russia is deteriorating due to the impact of natural and anthropogenic factors, such as global climate change, increasing water salinity, rising average annual temperatures, agricultural and industrial human activities, etc. Eutrophication of waters, particularly in the Taganrog Bay and the Azov Sea, causes degradation of both individual components of the ecosystem and entire communities of organisms. The collection and analysis of information on the current and predicted states of coastal systems becomes significant. Their variability over time, on a scale of up to several weeks, requires prompt forecasting of adverse phenomena, for which non-stationary spatially inhomogeneous interconnected mathematical models and effective numerical methods for their implementation have already been developed and continue to be improved, allowing us to “play” various scenarios of dynamic biological and geochemical processes in coastal systems.
Coastal systems’ state variability has become the object of many studies by Russian and foreign scientists in the field of biochemistry and biological kinetics. Diagnosis of the state and forecast of changes in the thermo-hydrodynamics of water bodies’ ecosystems is considered in a number of works [
1,
2]. The article [
3] presents estimates of the influence of changes in the hydrological regime on the species composition of phytoplankton and the habitats of the main species. The works [
4,
5] present the principles of modeling and monitoring the spatio-temporal dynamics of water bodies. A significant amount of work belongs to scientists from the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS) and the Shirshov Institute of Oceanology of Russian Academy of Sciences (IO RAS). Research is being conducted on the development of perspective procedures and algorithms for analyzing observational data (ensemble optimal interpolation, Kalman filters, three-dimensional and four-dimensional variational analysis) [
6,
7]. In works [
8,
9,
10] and others, biogeochemical transformations, hydrochemistry and rates of microbial processes in the water column, the assimilation of chemical elements by hydrobionts of various levels of trophic chains, and the influence of agriculture and industry on the state of the Black, Azov, and Caspian seas are studied. The hydrological regime, models of movement of the aquatic environment, biochemistry, biological kinetics, dynamics of primary bioproduction, and biogenic pollution of water bodies in the South of Russia were studied in [
11,
12]. The Marine Hydrophysical Institute of Russian Academy of Sciences (MGI RAS) is engaged in the creation and development of a model for diagnosing and forecasting the evolution of the main hydrophysical fields of the Black Sea, which operates in an operational mode (up to 5 days) [
13]. Russian and German scientists jointly developed a 1D model of biogeochemical processes [
14,
15]. This hydrophysical–biogeochemical model simulates the distribution of major nutrients (oxygen, nitrogen, sulfur, phosphorus, manganese, iron) in the pelagic redox layer for the Black and Baltic seas. An example of a three-dimensional model for modeling the processes of hydrodynamics, biological kinetics, and geochemical cycles in coastal systems is MARS [
16]. This model is integrated with models of sediment dynamics, microbiology and pollution distribution, and biogeochemical cycles. Cycles of nitrogen transformations in the presence of oxygen and in an anaerobic environment are considered in detail in [
17,
18]. Extensive studies of the influence of abiotic and biotic factors on phytoplankton communities are described in a number of works by Danish scientists. In [
19], the influence of a temperature increase over the course of a month on the development of populations of cyanobacteria and green algae in shallow lakes was studied. In this case, scenarios of low and high nutrient content, nitrogen in particular, were used. The work [
20] describes a 1D mathematical model for modeling hydrodynamic and biogeochemical processes in a shallow lake. The modeling of nitrogen cycles in marine systems, including the absorption of nitrogen by bottom sediments, is considered in the work of American scientists [
21]. At the same time, various approaches are used to describe denitrification processes: from mechanistic diagenetic models to empirical parameterizations of nitrogen fluxes across the sediment–water interface. A mathematical model of the hydrodynamics of coastal systems, which allows for the modeling of turbulent flows, is described in [
22]. The results of the study of the oscillatory behavior of solutions of some classes of differential equations, which allow the modeling of many processes in the ocean, are presented in [
23]. Numerical methods and difference schemes used to implement models of hydrodynamics and biogeochemistry are presented in [
24,
25,
26].
This paper presents a three-dimensional non-stationary model of the dynamics of plankton populations and geochemical cycles, which describes the change in the concentrations of the main biogenic substances (compounds of phosphorus, nitrogen and silicon, etc.), considering the mechanisms of plankton development, which allows for the modeling of geographical dynamics of plankton populations, the change in species composition due to changes in biogenic factors, and the development of adverse and dangerous phenomena (rapid flowering of poisonous species of phytoplankton and toxic algae, deadly phenomena). The initial-boundary value problem corresponding to the indicated model contains nonlinear terms, which significantly increases the computational complexity and the time required to solve it. To solve this type of problem, linearization methods are used such as, for example, Newton’s method or, applied to this problem, the method of introducing a time grid with values of nonlinear terms determined with a “delay”, and then solving a system of grid equations using well-established working mechanisms. An overview of modern linearization methods as well as their advantages and disadvantages are presented in [
27].
The authors performed a linearization of the initial-boundary value problem, i.e., a replacement of an approximate problem similar to it in terms of dynamic properties, the solution of which is quite close to the solution of the original problem. Linearization is carried out on a time grid; nonlinear coefficients are taken with a delay of one step of the time grid. Next, a problems chain, interrelated by initial conditions and the final solutions, linearized on a uniform time grid is built and, thus, the linearization of the 3D nonlinear model as a whole is carried out.
The use of linearization allows us to replace the functions included in the problem with linear ones and obtain a problem with a linear operator, which can later be solved using numerical methods.
Earlier, in the article [
28], the authors’ team presented a description of some aspects of the construction and numerical implementation of the model of biogeochemical cycles and biological kinetics of the multispecies population model. To this end, effective numerical methods and difference schemes have been developed that allow for the simulation of biogeochemical and hydrodynamic processes in a real computational domain of a complex shape [
24]. However, some issues of the analytical study of the model remained in the shadows. The authors attempted to close this gap.
It is well-known that the main assumption that allows us to go from a nonlinear problem to a linearized one is the assumption that the deviations of the values of the functions included in the consideration from their analogs, taken as initial ones during linearization, are small. Therefore, questions of “proximity” of solutions to the original nonlinear initial-boundary and linearized problems, on the basis of which the discrete model (difference scheme) was built, are of paramount importance. The scientific novelty of this work lies in the study of the convergence of a chain of linearized problems to the solution of the original nonlinear problems in the norm of the Hilbert space as the parameter, the step of the time grid on which the linearization was carried out, tends to zero. The originality of the work is determined not only by the results obtained, but also by the apparatus used, which is based on the methods of the theory of differential equations.
It is worth noting that the conditions for the existence and uniqueness of the solution of linearized problems were obtained earlier, along with the definition of requirements for the smoothness classes of the input data of the problem [
28].
3. Investigation of the Proximity of Solutions to the Linearized and Original Initial-Boundary Value Problems by the Energy Method
For the convenience of the study, we formulate the original (nonlinearized) system (1) as a chain of coupled initial-boundary value problems of the form:
where
,
,
,
, and
, with initial conditions:
as well as boundary conditions (3) and (4) considered on the interval
for Equation (9).
For brevity, we do not give the definition of functions
on each time interval
, considering that:
Now we consider the specifics of this problem: the specific respiration rate of each of the phytoplankton populations, given by a constant , such that .
This means that all the functions of the right-hand sides for the biogenic components () and are negative if we focus on the positiveness of all concentrations and , , and the type of functions and , and the typical values of the coefficients, also considering .
Naturally, such a conclusion is valid if there are no external sources of biogenic concentrations (for example, deficiently treated sewage effluents in a reservoir). We assume that there are initial values of the concentrations of biogenic components sufficient for the growth of phytoplankton populations in the time interval , where is the considered characteristic period for the entire system (for the Azov Sea, 7–8 months).
Now we turn to the analysis of the closeness of the solutions of the linearized and original problems, considering the assumptions made. We introduce the function:
We subtract from Equation (6) each
(number
) containing the equation of the number
containing
from the system of Equation (9),
. Considering the linearity of the operators participating in the equations, we obtain a system of the form:
where
,
,
, and
.
It is easy to see that the initial conditions are written as follows:
It should be noted that due to the presence of periodicity conditions:
However, this condition is not used in subsequent calculations.
The boundary conditions are formulated in an obvious way, based on the relation of (7) and (8), as well as that of (3) and (4):
We assume that each of the functions and is “square”-integrable in the domain for all , . We introduce the scalar product of functions and , such that for any , , there exist bounded integrals and , each of which is a continuously differentiable function of the variable .
The scalar product
is the expression:
which is a function depending on the variable
. Naturally, we assume that
.
Next, we introduce the Hilbert space
for functions
,
, …, square-integrable on
. We introduce the norm:
Obviously, each such norm is a non-negative function of the variable , continuously differentiable with respect to this variable. Further, for brevity, the product of differentials is denoted as .
Next, we multiply scalarly both parts of each
i-th equation by the function
and integrate over the variable
from
to
. We get:
We first transform the left side of the resulting equality, which we denote as
. We have:
Here, we take advantage of the fact that:
Using the Gauss formula (theorem), we transform the second term on the left side of equality (15) considering the boundary conditions (10)–(13). For convenience, we introduce the notation:
Next, we transform the expression
, included in the expression for
. We derive some generalization of Green’s first formula as applied to this expression. Then, we consider a vector function:
where
,
,
,
, and
,
.
We consider the expression
, which, in accordance with the Ostrogradsky–Gauss formula, can be written as the flow of a vector function
through the boundary surface
of the domain
:
where
is the normal component of vector
(in the direction of the outer normal to
) on the border
.
Considering the boundary conditions (10)–(13), it is easy to see that:
since the flow is zero on the remaining parts of the boundary
(relations (10), (11), (12)).
We then return to the expression
, which can be represented as:
From relations (18), (19), and (20), we have an equality, which can be considered the first Green’s formula in relation to our problem:
Substituting the right parts of equalities (16), (17), and (21)—the last one with a minus sign—into relation (15), we obtain:
Let
,
, and
be the maximum dimensions of the region
in the directions of the coordinate axes
,
, and
, respectively:
where
is the Euclidean distance function in
. Then, the Poincaré inequalities take place, which are further used to estimate the functional (22) and, as a result, to estimate
:
where
.
Considering Equations (14) and (22), as well as inequalities (23) and (24), we arrive at the following inequality:
In what follows, on the right side of the inequality, we omit the terms of the form:
Because of this, the right side of inequality (25) only increases. To complete the evaluation, we consider the expression:
For simplicity, we consider the case .
Based on equalities (2) and (5), we conclude that the value of this expression is maximum when there is no limit on the growth of the phytoplankton population by nutrients and abiogenic factors, that is, the population growth rate is the highest. Then, the value of the expression
is the largest and is also the largest (”worst”) estimate of the error. Based on the maximum value of the coefficient
, which is equal to the constant
, we come to the estimate:
We note that, in accordance with the hydrobiological meaning of the constants included in the expression in square brackets,
,
, and
and up to a certain point in time, when
, the concentration may increase. Considering inequalities (25) and (26), we obtain (for
),
We note that:
where
,
, and
.
Since all partial derivatives
and
are assumed to be continuous at the closure of the domain of definition, they are bounded; therefore:
Therefore, from (28) and (29), we have:
Then we have, considering (30):
where
.
Considering estimates (27) and (31), we have the inequality:
For convenience, we denote:
Then, estimate (32) can be written in a compact form as follows:
It is easy to distinguish (can be proved by induction) that the chain of inequalities–equalities is true:
We note that:
and the sum
is estimated as follows: It is obvious that
The final sum in parentheses on the right side of equality (35) is the sum of a finite number (
n terms) of a convergent geometric progression, in which the first term is 1 and the denominator is
. Then, the sum is written as:
where
. Considering (33), (34), and (36), we obtain the following estimate:
The last inequality implies an estimate that guarantees the closeness (convergence at
) of the solutions of the linearized and nonlinear problems for the substation
(initial function
) in
on the sequence of grids
, and the fulfillment in the case of any
inequality:
It should be noted that from the condition of periodicity and the chosen method of linearization of the problem, it follows that .
In addition, there is a (perhaps not the only) moment of time
,
, after which the decrease in the error begins. Non-uniqueness can be caused by the entry of biogenic components into area
from any source. By analogous reasoning, one can prove that:
where
and
are positive constants.